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Theorem funssfv 6923
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 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem funssfv
StepHypRef Expression
1 fvres 6921 . . . 4 (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹𝐴))
21eqcomd 2734 . . 3 (𝐴 ∈ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴))
3 funssres 6602 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43fveq1d 6904 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2790 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1107 1 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wss 3949  dom cdm 5682  cres 5684  Fun wfun 6547  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561
This theorem is referenced by:  fviunfun  7954  funelss  8057  funsssuppss  8201  frrlem10  8307  wfrlem12OLD  8347  wfrlem14OLD  8349  tfrlem9  8412  tfrlem11  8415  ac6sfi  9318  axdc3lem2  10482  axdc3lem4  10484  imasvscaval  17527  pserdv  26386  subgruhgredgd  29117  subumgredg2  29118  subupgr  29120  sspn  30566  bnj945  34437  bnj1502  34512  bnj545  34559  bnj548  34561  subfacp1lem2a  34823  subfacp1lem2b  34824  subfacp1lem5  34827  cvmliftlem10  34937  cvmliftlem13  34939
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