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Theorem svs3 25488
Description: A very concise definition of a subspace of a vector space. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
svs3  |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  | 
( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) } )
Distinct variable group:    x, v

Proof of Theorem svs3
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 resss 4979 . . . . . . . . 9  |-  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) 
C_  ( 2nd `  ( 2nd `  x ) )
3 sseq1 3199 . . . . . . . . 9  |-  ( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  ->  ( ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) )  <->  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) 
C_  ( 2nd `  ( 2nd `  x ) ) ) )
42, 3mpbiri 224 . . . . . . . 8  |-  ( ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) )  ->  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) )
5 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  v ) )
6 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  ( 2nd `  v
) )  =  ( 2nd `  ( 2nd `  v ) )
7 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  ran  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )
85, 6, 7vecax2 25454 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  Vec  ->  ( 2nd `  ( 2nd `  v
) ) : ( ran  ( 1st `  ( 1st `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) --> ran  ( 1st `  ( 2nd `  v ) ) )
9 fdm 5393 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  ( 2nd `  v ) ) : ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) --> ran  ( 1st `  ( 2nd `  v ) )  ->  dom  ( 2nd `  ( 2nd `  v
) )  =  ( ran  ( 1st `  ( 1st `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
109eqcomd 2288 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 2nd `  v ) ) : ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) ) --> ran  ( 1st `  ( 2nd `  v ) )  ->  ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) )
118, 10syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  Vec  ->  ( ran  ( 1st `  ( 1st `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  =  dom  ( 2nd `  ( 2nd `  v ) ) )
1211a1d 22 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  Vec  ->  (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) )
1312adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) )
14 fveq2 5525 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ( 1st `  ( 1st `  v
) )  =  ( 1st `  ( 1st `  x ) ) )
1514rneqd 4906 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  x ) ) )
1615eqcoms 2286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  x ) ) )
1716xpeq1d 4712 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ( ran  ( 1st `  ( 1st `  v ) )  X. 
ran  ( 1st `  ( 2nd `  v ) ) )  =  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
1817eqeq1d 2291 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ( ( ran  ( 1st `  ( 1st `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  =  dom  ( 2nd `  ( 2nd `  v ) )  <-> 
( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) )
1918imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ( (
( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  v
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) )  <->  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  ->  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) )  =  dom  ( 2nd `  ( 2nd `  v ) ) ) ) )
2013, 19syl5ib 210 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ( (
x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) ) )
2120eqcoms 2286 . . . . . . . . . . . . . 14  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ( (
x  e.  Vec  /\  v  e.  Vec  )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) ) )
2221impcom 419 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v
)  =  ( 1st `  x ) )  -> 
( ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  ->  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) ) )
2322imp 418 . . . . . . . . . . . 12  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  -> 
( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) )
2423adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  /\  ( 2nd `  ( 2nd `  v ) )  C_  ( 2nd `  ( 2nd `  x ) ) )  ->  ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  v ) ) )  =  dom  ( 2nd `  ( 2nd `  v
) ) )
2524reseq2d 4955 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  /\  ( 2nd `  ( 2nd `  v ) )  C_  ( 2nd `  ( 2nd `  x ) ) )  ->  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  dom  ( 2nd `  ( 2nd `  v
) ) ) )
26 eqid 2283 . . . . . . . . . . . . . 14  |-  ran  ( 1st `  ( 1st `  x
) )  =  ran  ( 1st `  ( 1st `  x ) )
27 eqid 2283 . . . . . . . . . . . . . 14  |-  ( 2nd `  ( 2nd `  x
) )  =  ( 2nd `  ( 2nd `  x ) )
28 eqid 2283 . . . . . . . . . . . . . 14  |-  ran  ( 1st `  ( 2nd `  x
) )  =  ran  ( 1st `  ( 2nd `  x ) )
2926, 27, 28vecax2 25454 . . . . . . . . . . . . 13  |-  ( x  e.  Vec  ->  ( 2nd `  ( 2nd `  x
) ) : ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) ) --> ran  ( 1st `  ( 2nd `  x ) ) )
30 ffun 5391 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 2nd `  x ) ) : ( ran  ( 1st `  ( 1st `  x
) )  X.  ran  ( 1st `  ( 2nd `  x ) ) ) --> ran  ( 1st `  ( 2nd `  x ) )  ->  Fun  ( 2nd `  ( 2nd `  x
) ) )
3129, 30syl 15 . . . . . . . . . . . 12  |-  ( x  e.  Vec  ->  Fun  ( 2nd `  ( 2nd `  x ) ) )
3231ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  ->  Fun  ( 2nd `  ( 2nd `  x ) ) )
33 funssres 5294 . . . . . . . . . . 11  |-  ( ( Fun  ( 2nd `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) ) 
C_  ( 2nd `  ( 2nd `  x ) ) )  ->  ( ( 2nd `  ( 2nd `  x
) )  |`  dom  ( 2nd `  ( 2nd `  v
) ) )  =  ( 2nd `  ( 2nd `  v ) ) )
3432, 33sylan 457 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  /\  ( 2nd `  ( 2nd `  v ) )  C_  ( 2nd `  ( 2nd `  x ) ) )  ->  ( ( 2nd `  ( 2nd `  x
) )  |`  dom  ( 2nd `  ( 2nd `  v
) ) )  =  ( 2nd `  ( 2nd `  v ) ) )
3525, 34eqtr2d 2316 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  /\  ( 2nd `  ( 2nd `  v ) )  C_  ( 2nd `  ( 2nd `  x ) ) )  ->  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) )
3635ex 423 . . . . . . . 8  |-  ( ( ( ( x  e. 
Vec  /\  v  e.  Vec  )  /\  ( 1st `  v )  =  ( 1st `  x
) )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) ) )  -> 
( ( 2nd `  ( 2nd `  v ) ) 
C_  ( 2nd `  ( 2nd `  x ) )  ->  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x
) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) ) )
374, 36impbid2 195 . . . . . . 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
) ) ) )  <-> 
( 2nd `  ( 2nd `  v ) ) 
C_  ( 2nd `  ( 2nd `  x ) ) ) )
3837pm5.32da 622 . . . . . 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 ) ) ) ) )  <->  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) ) )
3938pm5.32da 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
)  /\  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) ) ) )
40 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 ) ) ) ) ) ) )
41 3anass 938 . . . . 5  |-  ( ( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) )  <->  ( ( 1st `  v )  =  ( 1st `  x
)  /\  ( ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) ) )
4239, 40, 413bitr4g 279 . . . 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 )  /\  ( 1st `  ( 2nd `  v ) ) 
C_  ( 1st `  ( 2nd `  x ) )  /\  ( 2nd `  ( 2nd `  v ) ) 
C_  ( 2nd `  ( 2nd `  x ) ) ) ) )
4342rabbidva 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 )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) } )
4443mpteq2ia 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
)  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) } )
451, 44eqtri 2303 1  |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  | 
( ( 1st `  v
)  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v
) )  C_  ( 1st `  ( 2nd `  x
) )  /\  ( 2nd `  ( 2nd `  v
) )  C_  ( 2nd `  ( 2nd `  x
) ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249   -->wf 5251   ` cfv 5255   1stc1st 6120   2ndc2nd 6121    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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-vec 25450  df-svs 25486
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