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Theorem funssfv 6928
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 6926 . . . 4 (𝐴 ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐹𝐴))
21eqcomd 2741 . . 3 (𝐴 ∈ dom 𝐺 → (𝐹𝐴) = ((𝐹 ↾ dom 𝐺)‘𝐴))
3 funssres 6612 . . . 4 ((Fun 𝐹𝐺𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
43fveq1d 6909 . . 3 ((Fun 𝐹𝐺𝐹) → ((𝐹 ↾ dom 𝐺)‘𝐴) = (𝐺𝐴))
52, 4sylan9eqr 2797 . 2 (((Fun 𝐹𝐺𝐹) ∧ 𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
653impa 1109 1 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  dom cdm 5689  cres 5691  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  fviunfun  7968  funelss  8071  funsssuppss  8214  frrlem10  8319  wfrlem12OLD  8359  wfrlem14OLD  8361  tfrlem9  8424  tfrlem11  8427  ac6sfi  9318  axdc3lem2  10489  axdc3lem4  10491  imasvscaval  17585  pserdv  26488  subgruhgredgd  29316  subumgredg2  29317  subupgr  29319  sspn  30765  bnj945  34766  bnj1502  34841  bnj545  34888  bnj548  34890  subfacp1lem2a  35165  subfacp1lem2b  35166  subfacp1lem5  35169  cvmliftlem10  35279  cvmliftlem13  35281
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