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Theorem oprssov 6015
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 6013 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A ( F  |`  ( C  X.  D
) ) B )  =  ( A F B ) )
21adantl 277 . 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 5315 . . . . . . 7  |-  ( G  Fn  ( C  X.  D )  ->  dom  G  =  ( C  X.  D ) )
43reseq2d 4907 . . . . . 6  |-  ( G  Fn  ( C  X.  D )  ->  ( F  |`  dom  G )  =  ( F  |`  ( C  X.  D
) ) )
543ad2ant2 1019 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  ( F  |`  ( C  X.  D ) ) )
6 funssres 5258 . . . . . 6  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
763adant2 1016 . . . . 5  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |` 
dom  G )  =  G )
85, 7eqtr3d 2212 . . . 4  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( F  |`  ( C  X.  D
) )  =  G )
98oveqd 5891 . . 3  |-  ( ( Fun  F  /\  G  Fn  ( C  X.  D
)  /\  G  C_  F
)  ->  ( A
( F  |`  ( C  X.  D ) ) B )  =  ( A G B ) )
109adantr 276 . 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 2212 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3129    X. cxp 4624   dom cdm 4626    |` cres 4628   Fun wfun 5210    Fn wfn 5211  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877
This theorem is referenced by: (None)
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