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Theorem funssfv 5879
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 5878 . . . 4  |-  ( A  e.  dom  G  -> 
( ( F  |`  dom  G ) `  A
)  =  ( F `
 A ) )
21eqcomd 2475 . . 3  |-  ( A  e.  dom  G  -> 
( F `  A
)  =  ( ( F  |`  dom  G ) `
 A ) )
3 funssres 5626 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43fveq1d 5866 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
) `  A )  =  ( G `  A ) )
52, 4sylan9eqr 2530 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  e.  dom  G )  ->  ( F `  A )  =  ( G `  A ) )
653impa 1191 1  |-  ( ( Fun  F  /\  G  C_  F  /\  A  e. 
dom  G )  -> 
( F `  A
)  =  ( G `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   dom cdm 4999    |` cres 5001   Fun wfun 5580   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by:  funsssuppss  6923  tfrlem9  7051  tfrlem11  7054  ac6sfi  7760  axdc3lem2  8827  axdc3lem4  8829  imasvscaval  14786  pserdv  22555  eupap1  24649  sspn  25322  subfacp1lem2a  28261  subfacp1lem2b  28262  subfacp1lem5  28265  cvmliftlem10  28376  cvmliftlem13  28378  wfrlem12  28928  wfrlem14  28930  frrlem11  28973  bnj945  32911  bnj1502  32985  bnj545  33032  bnj548  33034
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