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Theorem oprssopvg 25034
Description: Value returned by the operation  G in terms of the value returned by the "super"-operation  F. (A version of oprssov 5989 adapted to partial operations.) (Contributed by FL, 5-Oct-2009.)
Assertion
Ref Expression
oprssopvg  |-  ( ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G
)  ->  ( A F B )  =  ( A G B ) )

Proof of Theorem oprssopvg
StepHypRef Expression
1 fvres 5542 . . . 4  |-  ( <. A ,  B >.  e. 
dom  G  ->  ( ( F  |`  dom  G ) `
 <. A ,  B >. )  =  ( F `
 <. A ,  B >. ) )
2 df-ov 5861 . . . 4  |-  ( A ( F  |`  dom  G
) B )  =  ( ( F  |`  dom  G ) `  <. A ,  B >. )
3 df-ov 5861 . . . 4  |-  ( A F B )  =  ( F `  <. A ,  B >. )
41, 2, 33eqtr4g 2340 . . 3  |-  ( <. A ,  B >.  e. 
dom  G  ->  ( A ( F  |`  dom  G
) B )  =  ( A F B ) )
543ad2ant3 978 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G
)  ->  ( A
( F  |`  dom  G
) B )  =  ( A F B ) )
6 funssres 5294 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
76oveqd 5875 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( A ( F  |`  dom  G ) B )  =  ( A G B ) )
873adant3 975 . 2  |-  ( ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G
)  ->  ( A
( F  |`  dom  G
) B )  =  ( A G B ) )
95, 8eqtr3d 2317 1  |-  ( ( Fun  F  /\  G  C_  F  /\  <. A ,  B >.  e.  dom  G
)  ->  ( A F B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643   dom cdm 4689    |` cres 4691   Fun wfun 5249   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  idsubfun  25858
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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