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Theorem prf2nd 14292
Description: Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prf1st.p  |-  P  =  ( F ⟨,⟩F  G )
prf1st.c  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
prf1st.d  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
Assertion
Ref Expression
prf2nd  |-  ( ph  ->  ( ( D  2ndF  E )  o.func 
P )  =  G )

Proof of Theorem prf2nd
Dummy variables  f  h  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . 7  |-  ( D  X.c  E )  =  ( D  X.c  E )
2 eqid 2435 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2435 . . . . . . . 8  |-  ( Base `  E )  =  (
Base `  E )
41, 2, 3xpcbas 14265 . . . . . . 7  |-  ( (
Base `  D )  X.  ( Base `  E
) )  =  (
Base `  ( D  X.c  E ) )
5 eqid 2435 . . . . . . 7  |-  (  Hom  `  ( D  X.c  E ) )  =  (  Hom  `  ( D  X.c  E ) )
6 prf1st.c . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 14050 . . . . . . . . . 10  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simprd 450 . . . . . . . 8  |-  ( ph  ->  D  e.  Cat )
109adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
11 prf1st.d . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( C 
Func  E ) )
12 funcrcl 14050 . . . . . . . . . 10  |-  ( G  e.  ( C  Func  E )  ->  ( C  e.  Cat  /\  E  e. 
Cat ) )
1311, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  ( C  e.  Cat  /\  E  e.  Cat )
)
1413simprd 450 . . . . . . . 8  |-  ( ph  ->  E  e.  Cat )
1514adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  E  e.  Cat )
16 eqid 2435 . . . . . . 7  |-  ( D  2ndF  E )  =  ( D  2ndF  E )
17 eqid 2435 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
18 relfunc 14049 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
19 1st2ndbr 6388 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 6, 19sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2117, 2, 20funcf1 14053 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
23 relfunc 14049 . . . . . . . . . . 11  |-  Rel  ( C  Func  E )
24 1st2ndbr 6388 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
2523, 11, 24sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  G
) ( C  Func  E ) ( 2nd `  G
) )
2617, 3, 25funcf1 14053 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  E ) )
2726ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)
28 opelxpi 4902 . . . . . . . 8  |-  ( ( ( ( 1st `  F
) `  x )  e.  ( Base `  D
)  /\  ( ( 1st `  G ) `  x )  e.  (
Base `  E )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
2922, 27, 28syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  <. ( ( 1st `  F ) `
 x ) ,  ( ( 1st `  G
) `  x ) >.  e.  ( ( Base `  D )  X.  ( Base `  E ) ) )
301, 4, 5, 10, 15, 16, 292ndf1 14282 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  2ndF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( 2nd `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
31 fvex 5734 . . . . . . 7  |-  ( ( 1st `  F ) `
 x )  e. 
_V
32 fvex 5734 . . . . . . 7  |-  ( ( 1st `  G ) `
 x )  e. 
_V
3331, 32op2nd 6348 . . . . . 6  |-  ( 2nd `  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  =  ( ( 1st `  G ) `
 x )
3430, 33syl6eq 2483 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  ( D  2ndF  E ) ) `  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. )  =  ( ( 1st `  G
) `  x )
)
3534mpteq2dva 4287 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( 1st `  ( D  2ndF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )  =  ( x  e.  (
Base `  C )  |->  ( ( 1st `  G
) `  x )
) )
36 prf1st.p . . . . . . 7  |-  P  =  ( F ⟨,⟩F  G )
37 eqid 2435 . . . . . . 7  |-  (  Hom  `  C )  =  (  Hom  `  C )
3836, 17, 37, 6, 11prfval 14286 . . . . . 6  |-  ( ph  ->  P  =  <. (
x  e.  ( Base `  C )  |->  <. (
( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >. )
39 fvex 5734 . . . . . . . 8  |-  ( Base `  C )  e.  _V
4039mptex 5958 . . . . . . 7  |-  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
)  e.  _V
4139, 39mpt2ex 6417 . . . . . . 7  |-  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) )  e.  _V
4240, 41op1std 6349 . . . . . 6  |-  ( P  =  <. ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( h  e.  ( x (  Hom  `  C ) y ) 
|->  <. ( ( x ( 2nd `  F
) y ) `  h ) ,  ( ( x ( 2nd `  G ) y ) `
 h ) >.
) ) >.  ->  ( 1st `  P )  =  ( x  e.  (
Base `  C )  |-> 
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
4338, 42syl 16 . . . . 5  |-  ( ph  ->  ( 1st `  P
)  =  ( x  e.  ( Base `  C
)  |->  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
) )
44 relfunc 14049 . . . . . . . 8  |-  Rel  (
( D  X.c  E ) 
Func  E )
451, 9, 14, 162ndfcl 14285 . . . . . . . 8  |-  ( ph  ->  ( D  2ndF  E )  e.  ( ( D  X.c  E
)  Func  E )
)
46 1st2ndbr 6388 . . . . . . . 8  |-  ( ( Rel  ( ( D  X.c  E )  Func  E
)  /\  ( D  2ndF  E )  e.  ( ( D  X.c  E )  Func  E
) )  ->  ( 1st `  ( D  2ndF  E ) ) ( ( D  X.c  E )  Func  E
) ( 2nd `  ( D  2ndF  E ) ) )
4744, 45, 46sylancr 645 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( D  2ndF  E ) ) ( ( D  X.c  E ) 
Func  E ) ( 2nd `  ( D  2ndF  E )
) )
484, 3, 47funcf1 14053 . . . . . 6  |-  ( ph  ->  ( 1st `  ( D  2ndF  E ) ) : ( ( Base `  D
)  X.  ( Base `  E ) ) --> (
Base `  E )
)
4948feqmptd 5771 . . . . 5  |-  ( ph  ->  ( 1st `  ( D  2ndF  E ) )  =  ( u  e.  ( ( Base `  D
)  X.  ( Base `  E ) )  |->  ( ( 1st `  ( D  2ndF  E ) ) `  u ) ) )
50 fveq2 5720 . . . . 5  |-  ( u  =  <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.  ->  ( ( 1st `  ( D  2ndF  E ) ) `  u )  =  ( ( 1st `  ( D  2ndF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) )
5129, 43, 49, 50fmptco 5893 . . . 4  |-  ( ph  ->  ( ( 1st `  ( D  2ndF  E ) )  o.  ( 1st `  P
) )  =  ( x  e.  ( Base `  C )  |->  ( ( 1st `  ( D  2ndF  E ) ) `  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. ) ) )
5226feqmptd 5771 . . . 4  |-  ( ph  ->  ( 1st `  G
)  =  ( x  e.  ( Base `  C
)  |->  ( ( 1st `  G ) `  x
) ) )
5335, 51, 523eqtr4d 2477 . . 3  |-  ( ph  ->  ( ( 1st `  ( D  2ndF  E ) )  o.  ( 1st `  P
) )  =  ( 1st `  G ) )
549ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  D  e.  Cat )
5514ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  E  e.  Cat )
56 relfunc 14049 . . . . . . . . . . . . . . . 16  |-  Rel  ( C  Func  ( D  X.c  E
) )
5736, 1, 6, 11prfcl 14290 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  P  e.  ( C 
Func  ( D  X.c  E
) ) )
58 1st2ndbr 6388 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( C  Func  ( D  X.c  E ) )  /\  P  e.  ( C  Func  ( D  X.c  E ) ) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
5956, 57, 58sylancr 645 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  P
) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
6017, 4, 59funcf1 14053 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  P
) : ( Base `  C ) --> ( (
Base `  D )  X.  ( Base `  E
) ) )
6160ffvelrnda 5862 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6261adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6362adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 1st `  P
) `  x )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6460ffvelrnda 5862 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  P ) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6564adantrl 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
6665adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 1st `  P
) `  y )  e.  ( ( Base `  D
)  X.  ( Base `  E ) ) )
671, 4, 5, 54, 55, 16, 63, 662ndf2 14283 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  =  ( 2nd  |`  ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) )
6867fveq1d 5722 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) )  =  ( ( 2nd  |`  (
( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
6959adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  P ) ( C  Func  ( D  X.c  E ) ) ( 2nd `  P ) )
70 simprl 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
71 simprr 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
7217, 37, 5, 69, 70, 71funcf2 14055 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  P
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )
7372ffvelrnda 5862 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  P ) y ) `  f
)  e.  ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )
74 fvres 5737 . . . . . . . . . 10  |-  ( ( ( x ( 2nd `  P ) y ) `
 f )  e.  ( ( ( 1st `  P ) `  x
) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
)  ->  ( ( 2nd  |`  ( ( ( 1st `  P ) `
 x ) (  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 2nd `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
7573, 74syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( 2nd  |`  (
( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) ) `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 2nd `  (
( x ( 2nd `  P ) y ) `
 f ) ) )
766ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  F  e.  ( C  Func  D ) )
7711ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  G  e.  ( C  Func  E ) )
7870adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  ->  x  e.  ( Base `  C ) )
7971adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
y  e.  ( Base `  C ) )
80 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
f  e.  ( x (  Hom  `  C
) y ) )
8136, 17, 37, 76, 77, 78, 79, 80prf2 14289 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( x ( 2nd `  P ) y ) `  f
)  =  <. (
( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. )
8281fveq2d 5724 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 2nd `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( 2nd `  <. ( ( x ( 2nd `  F ) y ) `
 f ) ,  ( ( x ( 2nd `  G ) y ) `  f
) >. ) )
83 fvex 5734 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  F
) y ) `  f )  e.  _V
84 fvex 5734 . . . . . . . . . . 11  |-  ( ( x ( 2nd `  G
) y ) `  f )  e.  _V
8583, 84op2nd 6348 . . . . . . . . . 10  |-  ( 2nd `  <. ( ( x ( 2nd `  F
) y ) `  f ) ,  ( ( x ( 2nd `  G ) y ) `
 f ) >.
)  =  ( ( x ( 2nd `  G
) y ) `  f )
8682, 85syl6eq 2483 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( 2nd `  (
( x ( 2nd `  P ) y ) `
 f ) )  =  ( ( x ( 2nd `  G
) y ) `  f ) )
8768, 75, 863eqtrd 2471 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
) ) )  /\  f  e.  ( x
(  Hom  `  C ) y ) )  -> 
( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) )  =  ( ( x ( 2nd `  G ) y ) `  f
) )
8887mpteq2dva 4287 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
f  e.  ( x (  Hom  `  C
) y )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( x ( 2nd `  G ) y ) `  f
) ) )
89 eqid 2435 . . . . . . . . 9  |-  (  Hom  `  E )  =  (  Hom  `  E )
9047adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  ( D  2ndF  E ) ) ( ( D  X.c  E )  Func  E
) ( 2nd `  ( D  2ndF  E ) ) )
914, 5, 89, 90, 62, 65funcf2 14055 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( 1st `  P
) `  x )
( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) : ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) --> ( ( ( 1st `  ( D  2ndF  E ) ) `  ( ( 1st `  P
) `  x )
) (  Hom  `  E
) ( ( 1st `  ( D  2ndF  E )
) `  ( ( 1st `  P ) `  y ) ) ) )
92 fcompt 5896 . . . . . . . 8  |-  ( ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) : ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) --> ( ( ( 1st `  ( D  2ndF  E ) ) `  ( ( 1st `  P
) `  x )
) (  Hom  `  E
) ( ( 1st `  ( D  2ndF  E )
) `  ( ( 1st `  P ) `  y ) ) )  /\  ( x ( 2nd `  P ) y ) : ( x (  Hom  `  C
) y ) --> ( ( ( 1st `  P
) `  x )
(  Hom  `  ( D  X.c  E ) ) ( ( 1st `  P
) `  y )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) ) )
9391, 72, 92syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( f  e.  ( x (  Hom  `  C ) y ) 
|->  ( ( ( ( 1st `  P ) `
 x ) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
) `  ( (
x ( 2nd `  P
) y ) `  f ) ) ) )
9425adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ( 1st `  G ) ( C  Func  E )
( 2nd `  G
) )
9517, 37, 89, 94, 70, 71funcf2 14055 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  G
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  G ) `  x
) (  Hom  `  E
) ( ( 1st `  G ) `  y
) ) )
9695feqmptd 5771 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
x ( 2nd `  G
) y )  =  ( f  e.  ( x (  Hom  `  C
) y )  |->  ( ( x ( 2nd `  G ) y ) `
 f ) ) )
9788, 93, 963eqtr4d 2477 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( x ( 2nd `  G ) y ) )
98973impb 1149 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) )  =  ( x ( 2nd `  G ) y ) )
9998mpt2eq3dva 6130 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  G ) y ) ) )
10017, 25funcfn2 14056 . . . . 5  |-  ( ph  ->  ( 2nd `  G
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
101 fnov 6170 . . . . 5  |-  ( ( 2nd `  G )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  G
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  G ) y ) ) )
102100, 101sylib 189 . . . 4  |-  ( ph  ->  ( 2nd `  G
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  G ) y ) ) )
10399, 102eqtr4d 2470 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  P ) `  x
) ( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) )  =  ( 2nd `  G ) )
10453, 103opeq12d 3984 . 2  |-  ( ph  -> 
<. ( ( 1st `  ( D  2ndF  E ) )  o.  ( 1st `  P
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) ) >.  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
10517, 57, 45cofuval 14069 . 2  |-  ( ph  ->  ( ( D  2ndF  E )  o.func 
P )  =  <. ( ( 1st `  ( D  2ndF  E ) )  o.  ( 1st `  P
) ) ,  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  P
) `  x )
( 2nd `  ( D  2ndF  E ) ) ( ( 1st `  P
) `  y )
)  o.  ( x ( 2nd `  P
) y ) ) ) >. )
106 1st2nd 6385 . . 3  |-  ( ( Rel  ( C  Func  E )  /\  G  e.  ( C  Func  E
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
10723, 11, 106sylancr 645 . 2  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
108104, 105, 1073eqtr4d 2477 1  |-  ( ph  ->  ( ( D  2ndF  E )  o.func 
P )  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    e. cmpt 4258    X. cxp 4868    |` cres 4872    o. ccom 4874   Rel wrel 4875    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13459    Hom chom 13530   Catccat 13879    Func cfunc 14041    o.func ccofu 14043    X.c cxpc 14255    2ndF c2ndf 14257   ⟨,⟩F cprf 14258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-hom 13543  df-cco 13544  df-cat 13883  df-cid 13884  df-func 14045  df-cofu 14047  df-xpc 14259  df-2ndf 14261  df-prf 14262
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