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Theorem svs2 25487
Description: A textbook definition. A sub-vector space of a vector space 
x is a subset that is itself a vector space under the inherited operations. (Contributed by FL, 31-Dec-2010.)
Assertion
Ref Expression
svs2  |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  | 
( ( 1st `  v
)  =  ( 1st `  x )  /\  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) } )
Distinct variable group:    x, v, w

Proof of Theorem svs2
StepHypRef Expression
1 df-svs 25486 . 2  |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  | 
( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) } )
2 3anass 938 . . . . 5  |-  ( ( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  <->  ( ( 1st `  v )  =  ( 1st `  x
)  /\  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
3 rnss 4907 . . . . . . . . . . 11  |-  ( ( 1st `  ( 2nd `  v ) )  C_  ( 1st `  ( 2nd `  x ) )  ->  ran  ( 1st `  ( 2nd `  v ) ) 
C_  ran  ( 1st `  ( 2nd `  x
) ) )
4 fvex 5539 . . . . . . . . . . . . 13  |-  ( 1st `  ( 2nd `  v
) )  e.  _V
54rnex 4942 . . . . . . . . . . . 12  |-  ran  ( 1st `  ( 2nd `  v
) )  e.  _V
65elpw 3631 . . . . . . . . . . 11  |-  ( ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  <->  ran  ( 1st `  ( 2nd `  v
) )  C_  ran  ( 1st `  ( 2nd `  x ) ) )
73, 6sylibr 203 . . . . . . . . . 10  |-  ( ( 1st `  ( 2nd `  v ) )  C_  ( 1st `  ( 2nd `  x ) )  ->  ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )
8 eqid 2283 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1st `  ( 2nd `  x
) )  =  ( 1st `  ( 2nd `  x ) )
98vecax1 25453 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  Vec  ->  ( 1st `  ( 2nd `  x
) )  e.  AbelOp )
10 ablogrpo 20951 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
AbelOp  ->  ( 1st `  ( 2nd `  x ) )  e.  GrpOp )
119, 10syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  Vec  ->  ( 1st `  ( 2nd `  x
) )  e.  GrpOp )
12 eqid 2283 . . . . . . . . . . . . . . . . . . . . . 22  |-  ran  ( 1st `  ( 2nd `  x
) )  =  ran  ( 1st `  ( 2nd `  x ) )
1312grpofo 20866 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
GrpOp  ->  ( 1st `  ( 2nd `  x ) ) : ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )
-onto->
ran  ( 1st `  ( 2nd `  x ) ) )
14 fofun 5452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  ( 2nd `  x ) ) : ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )
-onto->
ran  ( 1st `  ( 2nd `  x ) )  ->  Fun  ( 1st `  ( 2nd `  x
) ) )
1513, 14syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
GrpOp  ->  Fun  ( 1st `  ( 2nd `  x
) ) )
16 funssres 5294 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  ( 1st `  ( 2nd `  x ) )  /\  ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) ) )  ->  ( ( 1st `  ( 2nd `  x
) )  |`  dom  ( 1st `  ( 2nd `  v
) ) )  =  ( 1st `  ( 2nd `  v ) ) )
1716ex 423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Fun  ( 1st `  ( 2nd `  x ) )  ->  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  ->  (
( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) ) ) )
18 eqid 2283 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1st `  ( 2nd `  v
) )  =  ( 1st `  ( 2nd `  v ) )
1918vecax1 25453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( v  e.  Vec  ->  ( 1st `  ( 2nd `  v
) )  e.  AbelOp )
20 ablogrpo 20951 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 1st `  ( 2nd `  v ) )  e. 
AbelOp  ->  ( 1st `  ( 2nd `  v ) )  e.  GrpOp )
2119, 20syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( v  e.  Vec  ->  ( 1st `  ( 2nd `  v
) )  e.  GrpOp )
22 dmrngrp 25339 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 1st `  ( 2nd `  v ) )  e. 
GrpOp  ->  dom  ( 1st `  ( 2nd `  v
) )  =  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
2322reseq2d 4955 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 1st `  ( 2nd `  v ) )  e. 
GrpOp  ->  ( ( 1st `  ( 2nd `  x
) )  |`  dom  ( 1st `  ( 2nd `  v
) ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
24 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  /\  ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) ) )  -> 
( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  =  ( 1st `  ( 2nd `  v
) ) )
2524eqcomd 2288 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  /\  ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) ) )  -> 
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )
2625ex 423 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  ->  ( (
( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
2726eqcoms 2286 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  ->  ( ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
2823, 27syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  ( 2nd `  v ) )  e. 
GrpOp  ->  ( ( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
2921, 28syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( v  e.  Vec  ->  (
( ( 1st `  ( 2nd `  x ) )  |`  dom  ( 1st `  ( 2nd `  v ) ) )  =  ( 1st `  ( 2nd `  v
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
3017, 29syl9 66 . . . . . . . . . . . . . . . . . . . 20  |-  ( Fun  ( 1st `  ( 2nd `  x ) )  ->  ( v  e. 
Vec  ->  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
3115, 30syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
GrpOp  ->  ( v  e. 
Vec  ->  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
3211, 31syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  Vec  ->  (
v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) )
3332imp 418 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
3433adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
3534imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  -> 
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )
36353adant3 975 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  ->  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
37 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  ->  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
3836, 37jca 518 . . . . . . . . . . . . 13  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
39 xpeq12 4708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( w  =  ran  ( 1st `  ( 2nd `  v
) )  /\  w  =  ran  ( 1st `  ( 2nd `  v ) ) )  ->  ( w  X.  w )  =  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
4039anidms 626 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
w  X.  w )  =  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )
4140reseq2d 4955 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
4241eqeq2d 2294 . . . . . . . . . . . . . . . 16  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  <->  ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
43 xpeq2 4704 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  ( ran  ( 1st `  ( 1st `  x ) )  X.  w )  =  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )
4443reseq2d 4955 . . . . . . . . . . . . . . . . 17  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
4544eqeq2d 2294 . . . . . . . . . . . . . . . 16  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) )  <-> 
( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
4642, 45anbi12d 691 . . . . . . . . . . . . . . 15  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  <->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
4746rspcev 2884 . . . . . . . . . . . . . 14  |-  ( ( ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  /\  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) )
4847ex 423 . . . . . . . . . . . . 13  |-  ( ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) )
4938, 48syl5com 26 . . . . . . . . . . . 12  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  ->  ( ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) )
50493exp 1150 . . . . . . . . . . 11  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  ->  ( ran  ( 1st `  ( 2nd `  v ) )  e. 
~P ran  ( 1st `  ( 2nd `  x
) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) ) ) )
5150com14 82 . . . . . . . . . 10  |-  ( ran  ( 1st `  ( 2nd `  v ) )  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  ->  ( (
( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x ) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) ) ) )
527, 51mpcom 32 . . . . . . . . 9  |-  ( ( 1st `  ( 2nd `  v ) )  C_  ( 1st `  ( 2nd `  x ) )  -> 
( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  ->  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) ) )
5352imp 418 . . . . . . . 8  |-  ( ( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )  ->  ( (
( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x ) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) )
5453com12 27 . . . . . . 7  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  ->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) )
55 resss 4979 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  C_  ( 1st `  ( 2nd `  x
) )
56 sseq1 3199 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  <-> 
( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  C_  ( 1st `  ( 2nd `  x
) ) ) )
5755, 56mpbiri 224 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )
5857adantr 451 . . . . . . . . . . . 12  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) ) )
59583ad2ant2 977 . . . . . . . . . . 11  |-  ( ( w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  /\  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  /\  ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) ) )  -> 
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) ) )
60 dmeq 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( dom  ( 1st `  ( 2nd `  v ) )  =  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  ->  dom  dom  ( 1st `  ( 2nd `  v
) )  =  dom  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
61 dmxpid 4898 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  dom  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  =  ran  ( 1st `  ( 2nd `  v ) )
62 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( dom  dom  ( 1st `  ( 2nd `  v
) )  =  dom  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  /\  dom  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  ->  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )
63 dmrngrp 25339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
GrpOp  ->  dom  ( 1st `  ( 2nd `  x
) )  =  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) ) )
64 vex 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  w  e. 
_V
6564elpw 3631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  <->  w  C_  ran  ( 1st `  ( 2nd `  x ) ) )
66 xpss12 4792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( w  C_  ran  ( 1st `  ( 2nd `  x
) )  /\  w  C_ 
ran  ( 1st `  ( 2nd `  x ) ) )  ->  ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) ) )
67 sseq2 3200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  =  dom  ( 1st `  ( 2nd `  x ) )  ->  ( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  <->  ( w  X.  w )  C_  dom  ( 1st `  ( 2nd `  x ) ) ) )
68 ssdmres 4977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( ( w  X.  w ) 
C_  dom  ( 1st `  ( 2nd `  x
) )  <->  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ( w  X.  w ) )
69 dmeq 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44  |-  ( dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ( w  X.  w )  ->  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  dom  ( w  X.  w
) )
70 dmxpid 4898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44  |-  dom  (
w  X.  w )  =  w
71 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45  |-  ( ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  dom  (
w  X.  w )  /\  dom  ( w  X.  w )  =  w )  ->  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  w )
72 dmeq 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  dom  ( 1st `  ( 2nd `  v ) )  =  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) ) )
73 dmeq 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48  |-  ( dom  ( 1st `  ( 2nd `  v ) )  =  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  dom  dom  ( 1st `  ( 2nd `  v ) )  =  dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )
7472, 73syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  dom  dom  ( 1st `  ( 2nd `  v ) )  =  dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )
75 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50  |-  ( ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  dom  dom  ( 1st `  ( 2nd `  v ) )  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  ->  dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )
76 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54  |-  ( ( w  =  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ran  ( 1st `  ( 2nd `  v ) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) )
77 idd 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56  |-  ( v  e.  Vec  ->  (
w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) )
7877a1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ( v  e.  Vec  ->  ( w  =  ran  ( 1st `  ( 2nd `  v ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
7978com13 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54  |-  ( w  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
8076, 79syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53  |-  ( ( w  =  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ran  ( 1st `  ( 2nd `  v ) ) )  ->  ( v  e. 
Vec  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
8180ex 423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52  |-  ( w  =  dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
8281eqcoms 2286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51  |-  ( dom 
dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  w  -> 
( dom  dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( v  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
8382com3l 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50  |-  ( dom 
dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  w  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
8475, 83syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49  |-  ( ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  dom  dom  ( 1st `  ( 2nd `  v ) )  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  -> 
( v  e.  Vec  ->  ( dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  w  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
8584ex 423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48  |-  ( dom 
dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  dom  dom  ( 1st `  ( 2nd `  v ) )  -> 
( dom  dom  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( v  e.  Vec  ->  ( dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  w  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
8685eqcoms 2286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47  |-  ( dom 
dom  ( 1st `  ( 2nd `  v ) )  =  dom  dom  (
( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  w  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
8774, 86syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  w  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
8887com14 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45  |-  ( dom 
dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  w  -> 
( dom  dom  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
8971, 88syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44  |-  ( ( dom  dom  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  =  dom  (
w  X.  w )  /\  dom  ( w  X.  w )  =  w )  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
9069, 70, 89sylancl 643 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  |-  ( dom  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  =  ( w  X.  w )  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v ) )  ->  ( v  e. 
Vec  ->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
( 1st `  v
)  =  ( 1st `  x )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
9168, 90sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42  |-  ( ( w  X.  w ) 
C_  dom  ( 1st `  ( 2nd `  x
) )  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
9267, 91syl6bi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  |-  ( ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  =  dom  ( 1st `  ( 2nd `  x ) )  ->  ( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( dom  dom  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
9392eqcoms 2286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  ->  ( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( dom  dom  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( v  e.  Vec  ->  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
9493com24 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  ->  ( v  e. 
Vec  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v ) )  ->  ( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
95943imp 1145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  /\  v  e.  Vec  /\ 
dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  -> 
( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( ( 1st `  v )  =  ( 1st `  x
)  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
9695com24 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  /\  v  e.  Vec  /\ 
dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( (
w  X.  w ) 
C_  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
97963imp 1145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( ( dom  ( 1st `  ( 2nd `  x
) )  =  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  /\  v  e.  Vec  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  /\  ( 1st `  v )  =  ( 1st `  x
)  /\  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )  -> 
( ( w  X.  w )  C_  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) )
9866, 97syl5com 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( w  C_  ran  ( 1st `  ( 2nd `  x
) )  /\  w  C_ 
ran  ( 1st `  ( 2nd `  x ) ) )  ->  ( (
( dom  ( 1st `  ( 2nd `  x
) )  =  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  /\  v  e.  Vec  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  /\  ( 1st `  v )  =  ( 1st `  x
)  /\  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) )
9965, 65, 98sylancb 646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( ( dom  ( 1st `  ( 2nd `  x
) )  =  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  /\  v  e.  Vec  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  /\  ( 1st `  v )  =  ( 1st `  x
)  /\  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) )
10099com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( dom  ( 1st `  ( 2nd `  x
) )  =  ( ran  ( 1st `  ( 2nd `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) )  /\  v  e.  Vec  /\  dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  /\  ( 1st `  v )  =  ( 1st `  x
)  /\  ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) ) )  -> 
( w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) )
1011003exp 1150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  /\  v  e.  Vec  /\ 
dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
1021013exp 1150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( dom  ( 1st `  ( 2nd `  x ) )  =  ( ran  ( 1st `  ( 2nd `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) )  ->  ( v  e. 
Vec  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v ) )  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
10363, 102syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( 1st `  ( 2nd `  x ) )  e. 
GrpOp  ->  ( v  e. 
Vec  ->  ( dom  dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v ) )  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
10411, 103syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( x  e.  Vec  ->  (
v  e.  Vec  ->  ( dom  dom  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
105104com13 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( dom 
dom  ( 1st `  ( 2nd `  v ) )  =  ran  ( 1st `  ( 2nd `  v
) )  ->  (
v  e.  Vec  ->  ( x  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
10662, 105syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( dom  dom  ( 1st `  ( 2nd `  v
) )  =  dom  ( ran  ( 1st `  ( 2nd `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  /\  dom  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  ran  ( 1st `  ( 2nd `  v
) ) )  -> 
( v  e.  Vec  ->  ( x  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
10760, 61, 106sylancl 643 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( dom  ( 1st `  ( 2nd `  v ) )  =  ( ran  ( 1st `  ( 2nd `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  ->  ( v  e. 
Vec  ->  ( x  e. 
Vec  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
10822, 107syl 15 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 1st `  ( 2nd `  v ) )  e. 
GrpOp  ->  ( v  e. 
Vec  ->  ( x  e. 
Vec  ->  ( ( 1st `  v )  =  ( 1st `  x )  ->  ( ( 1st `  ( 2nd `  v
) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
10921, 108mpcom 32 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( v  e.  Vec  ->  (
x  e.  Vec  ->  ( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
110109impcom 419 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( 1st `  v
)  =  ( 1st `  x )  ->  (
( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )
111110imp 418 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  ->  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  w  =  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
112111com12 27 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) ) ) )
1131123imp 1145 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  w  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )  ->  w  =  ran  ( 1st `  ( 2nd `  v
) ) )
114113, 43syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  w  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ran  ( 1st `  ( 1st `  x
) )  X.  w
)  =  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
115114reseq2d 4955 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  w  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )
116115eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  w  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) )  <-> 
( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
117116biimpd 198 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  w  e.  ~P ran  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) )  ->  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
1181173exp 1150 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  (
w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  -> 
( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) )  ->  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
119118com24 81 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  ->  (
( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) )  ->  ( w  e. 
~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) ) )
120119imp 418 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  ->  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
121120com12 27 . . . . . . . . . . . 12  |-  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) ) )
1221213imp 1145 . . . . . . . . . . 11  |-  ( ( w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  /\  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  /\  ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) ) )  -> 
( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )
12359, 122jca 518 . . . . . . . . . 10  |-  ( ( w  e.  ~P ran  ( 1st `  ( 2nd `  x ) )  /\  ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  /\  ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) ) )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
1241233exp 1150 . . . . . . . . 9  |-  ( w  e.  ~P ran  ( 1st `  ( 2nd `  x
) )  ->  (
( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) ) )
125124rexlimiv 2661 . . . . . . . 8  |-  ( E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x ) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x
) )  |`  (
w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  ->  (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) )
126125com12 27 . . . . . . 7  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( E. w  e. 
~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) ) )
12754, 126impbid 183 . . . . . 6  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) )  <->  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) )
128127pm5.32da 622 . . . . 5  |-  ( ( x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( ( 1st `  v )  =  ( 1st `  x )  /\  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) )  <->  ( ( 1st `  v )  =  ( 1st `  x
)  /\  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) ) )
1292, 128syl5bb 248 . . . 4  |-  ( ( x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( ( 1st `  v )  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )  <->  ( ( 1st `  v )  =  ( 1st `  x )  /\  E. w  e. 
~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) ) )
130129rabbidva 2779 . . 3  |-  ( x  e.  Vec  ->  { v  e.  Vec  |  ( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) }  =  { v  e.  Vec  |  ( ( 1st `  v
)  =  ( 1st `  x )  /\  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) } )
131130mpteq2ia 4102 . 2  |-  ( x  e.  Vec  |->  { v  e.  Vec  |  ( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) ) ) } )  =  ( x  e. 
Vec  |->  { v  e. 
Vec  |  ( ( 1st `  v )  =  ( 1st `  x
)  /\  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) } )
1321, 131eqtri 2303 1  |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  | 
( ( 1st `  v
)  =  ( 1st `  x )  /\  E. w  e.  ~P  ran  ( 1st `  ( 2nd `  x
) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w
) )  /\  ( 2nd `  ( 2nd `  v
) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  w
) ) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853   AbelOpcablo 20948    Vec cvec 25449   SubVeccsvec 25485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-vec 25450  df-svs 25486
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