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Theorem oprssov 6217
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 6215 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )
21adantl 454 . 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 5546 . . . . . . 7  |-  ( G  Fn  ( C  X.  D )  ->  dom  G  =  ( C  X.  D ) )
43reseq2d 5148 . . . . . 6  |-  ( G  Fn  ( C  X.  D )  ->  ( F  |`  dom  G )  =  ( F  |`  ( C  X.  D
) ) )
543ad2ant2 980 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  ( F  |`  ( C  X.  D ) ) )
6 funssres 5495 . . . . . 6  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
763adant2 977 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  G )
85, 7eqtr3d 2472 . . . 4  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |`  ( C  X.  D
) )  =  G )
98oveqd 6100 . . 3  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A G B ) )
109adantr 453 . 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 2472 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322    X. cxp 4878   dom cdm 4880    |` cres 4882   Fun wfun 5450    Fn wfn 5451  (class class class)co 6083
This theorem is referenced by:  sspg  22229  ssps  22231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-ov 6086
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