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Theorem svs3 25591
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 25589 . 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 4995 . . . . . . . . 9  |-  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) ) 
C_  ( 2nd `  ( 2nd `  x ) )
3 sseq1 3212 . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  v ) )
6 eqid 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  ( 2nd `  v
) )  =  ( 2nd `  ( 2nd `  v ) )
7 eqid 2296 . . . . . . . . . . . . . . . . . . . 20  |-  ran  ( 1st `  ( 2nd `  v
) )  =  ran  ( 1st `  ( 2nd `  v ) )
85, 6, 7vecax2 25557 . . . . . . . . . . . . . . . . . . 19  |-  ( v  e.  Vec  ->  ( 2nd `  ( 2nd `  v
) ) : ( ran  ( 1st `  ( 1st `  v ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) --> ran  ( 1st `  ( 2nd `  v ) ) )
9 fdm 5409 . . . . . . . . . . . . . . . . . . . 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 2301 . . . . . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ( 1st `  ( 1st `  v
) )  =  ( 1st `  ( 1st `  x ) ) )
1514rneqd 4922 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  v )  =  ( 1st `  x
)  ->  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  x ) ) )
1615eqcoms 2299 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ran  ( 1st `  ( 1st `  v
) )  =  ran  ( 1st `  ( 1st `  x ) ) )
1716xpeq1d 4728 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  x )  =  ( 1st `  v
)  ->  ( ran  ( 1st `  ( 1st `  v ) )  X. 
ran  ( 1st `  ( 2nd `  v ) ) )  =  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
) ) ) )
1817eqeq1d 2304 . . . . . . . . . . . . . . . . 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 2299 . . . . . . . . . . . . . 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 4971 . . . . . . . . . 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 2296 . . . . . . . . . . . . . 14  |-  ran  ( 1st `  ( 1st `  x
) )  =  ran  ( 1st `  ( 1st `  x ) )
27 eqid 2296 . . . . . . . . . . . . . 14  |-  ( 2nd `  ( 2nd `  x
) )  =  ( 2nd `  ( 2nd `  x ) )
28 eqid 2296 . . . . . . . . . . . . . 14  |-  ran  ( 1st `  ( 2nd `  x
) )  =  ran  ( 1st `  ( 2nd `  x ) )
2926, 27, 28vecax2 25557 . . . . . . . . . . . . 13  |-  ( x  e.  Vec  ->  ( 2nd `  ( 2nd `  x
) ) : ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  x
) ) ) --> ran  ( 1st `  ( 2nd `  x ) ) )
30 ffun 5407 . . . . . . . . . . . . 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 5310 . . . . . . . . . . 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 2329 . . . . . . . . 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 2792 . . 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 4118 . 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 2316 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 1632    e. wcel 1696   {crab 2560    C_ wss 3165    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   Fun wfun 5265   -->wf 5267   ` cfv 5271   1stc1st 6136   2ndc2nd 6137    Vec cvec 25552   SubVeccsvec 25588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-vec 25553  df-svs 25589
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