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Theorem funssfv 5885
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 5884 . . . 4  |-  ( A  e.  dom  G  -> 
( ( F  |`  dom  G ) `  A
)  =  ( F `
 A ) )
21eqcomd 2459 . . 3  |-  ( A  e.  dom  G  -> 
( F `  A
)  =  ( ( F  |`  dom  G ) `
 A ) )
3 funssres 5625 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
43fveq1d 5872 . . 3  |-  ( ( Fun  F  /\  G  C_  F )  ->  (
( F  |`  dom  G
) `  A )  =  ( G `  A ) )
52, 4sylan9eqr 2509 . 2  |-  ( ( ( Fun  F  /\  G  C_  F )  /\  A  e.  dom  G )  ->  ( F `  A )  =  ( G `  A ) )
653impa 1204 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 371    /\ w3a 986    = wceq 1446    e. wcel 1889    C_ wss 3406   dom cdm 4837    |` cres 4839   Fun wfun 5579   ` cfv 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5549  df-fun 5587  df-fv 5593
This theorem is referenced by:  funsssuppss  6946  wfrlem12  7052  wfrlem14  7054  tfrlem9  7108  tfrlem11  7111  ac6sfi  7820  axdc3lem2  8886  axdc3lem4  8888  imasvscaval  15456  pserdv  23396  eupap1  25716  sspn  26387  bnj945  29597  bnj1502  29671  bnj545  29718  bnj548  29720  subfacp1lem2a  29915  subfacp1lem2b  29916  subfacp1lem5  29919  cvmliftlem10  30029  cvmliftlem13  30031  frrlem11  30538  subgruhgredgd  39366  subumgredg2  39367  subupgr  39369
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