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Theorem funssfv 5543
Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
Assertion
Ref Expression
funssfv  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )

Proof of Theorem funssfv
StepHypRef Expression
1 fvres 5542 . . . 4  |-  ( A  e.  dom  G  -> 
( ( F  |`  dom  G ) `  A
)  =  ( F `
 A ) )
21eqcomd 2288 . . 3  |-  ( A  e.  dom  G  -> 
( F `  A
)  =  ( ( F  |`  dom  G ) `
 A ) )
3 funssres 5294 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43fveq1d 5527 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
) `  A )  =  ( G `  A ) )
52, 4sylan9eqr 2337 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  e.  dom  G )  ->  ( F `  A )  =  ( G `  A ) )
653impa 1146 1  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   dom cdm 4689    |` cres 4691   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  tfrlem9  6401  tfrlem11  6404  ac6sfi  7101  axdc3lem2  8077  axdc3lem4  8079  imasvscaval  13440  pserdv  19805  sspn  21312  subfacp1lem2a  23122  subfacp1lem2b  23123  subfacp1lem5  23126  cvmliftlem10  23236  cvmliftlem13  23238  eupap1  23311  wfrlem12  23678  wfrlem14  23680  frrlem11  23704  prl2  24581  idsubidsup  25269  idsubfun  25270  bnj945  28178  bnj1502  28253  bnj545  28300  bnj548  28302
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
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