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Theorem funssfv 6927
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 6925 . . . 4 (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹𝐴))
21eqcomd 2743 . . 3 (𝐴 ∈ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴))
3 funssres 6610 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43fveq1d 6908 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2799 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1110 1 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  dom cdm 5685  cres 5687  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  fviunfun  7969  funelss  8072  funsssuppss  8215  frrlem10  8320  wfrlem12OLD  8360  wfrlem14OLD  8362  tfrlem9  8425  tfrlem11  8428  ac6sfi  9320  axdc3lem2  10491  axdc3lem4  10493  imasvscaval  17583  pserdv  26473  subgruhgredgd  29301  subumgredg2  29302  subupgr  29304  sspn  30755  bnj945  34787  bnj1502  34862  bnj545  34909  bnj548  34911  subfacp1lem2a  35185  subfacp1lem2b  35186  subfacp1lem5  35189  cvmliftlem10  35299  cvmliftlem13  35301
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