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Theorem funssfv 5705
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 5704 . . . 4  |-  ( A  e.  dom  G  -> 
( ( F  |`  dom  G ) `  A
)  =  ( F `
 A ) )
21eqcomd 2409 . . 3  |-  ( A  e.  dom  G  -> 
( F `  A
)  =  ( ( F  |`  dom  G ) `
 A ) )
3 funssres 5452 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43fveq1d 5689 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
) `  A )  =  ( G `  A ) )
52, 4sylan9eqr 2458 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  e.  dom  G )  ->  ( F `  A )  =  ( G `  A ) )
653impa 1148 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3280   dom cdm 4837    |` cres 4839   Fun wfun 5407   ` cfv 5413
This theorem is referenced by:  tfrlem9  6605  tfrlem11  6608  ac6sfi  7310  axdc3lem2  8287  axdc3lem4  8289  imasvscaval  13718  pserdv  20298  eupap1  21651  sspn  22188  subfacp1lem2a  24819  subfacp1lem2b  24820  subfacp1lem5  24823  cvmliftlem10  24934  cvmliftlem13  24936  wfrlem12  25481  wfrlem14  25483  frrlem11  25507  bnj945  28850  bnj1502  28925  bnj545  28972  bnj548  28974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421
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