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Theorem oprssov 6438
Description: The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.)
Assertion
Ref Expression
oprssov  |-  ( ( ( Fun  F  /\  G  Fn  ( C  X.  D )  /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( A F B )  =  ( A G B ) )

Proof of Theorem oprssov
StepHypRef Expression
1 ovres 6436 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )
21adantl 468 . 2  |-  ( ( ( Fun  F  /\  G  Fn  ( C  X.  D )  /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A F B ) )
3 fndm 5675 . . . . . . 7  |-  ( G  Fn  ( C  X.  D )  ->  dom  G  =  ( C  X.  D ) )
43reseq2d 5105 . . . . . 6  |-  ( G  Fn  ( C  X.  D )  ->  ( F  |`  dom  G )  =  ( F  |`  ( C  X.  D
) ) )
543ad2ant2 1030 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  ( F  |`  ( C  X.  D ) ) )
6 funssres 5622 . . . . . 6  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
763adant2 1027 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  G )
85, 7eqtr3d 2487 . . . 4  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |`  ( C  X.  D
) )  =  G )
98oveqd 6307 . . 3  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A G B ) )
109adantr 467 . 2  |-  ( ( ( Fun  F  /\  G  Fn  ( C  X.  D )  /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A G B ) )
112, 10eqtr3d 2487 1  |-  ( ( ( Fun  F  /\  G  Fn  ( C  X.  D )  /\  G  C_  F )  /\  ( A  e.  C  /\  B  e.  D )
)  ->  ( A F B )  =  ( A G B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    C_ wss 3404    X. cxp 4832   dom cdm 4834    |` cres 4836   Fun wfun 5576    Fn wfn 5577  (class class class)co 6290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-ov 6293
This theorem is referenced by:  sspg  26367  ssps  26369
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