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Theorem fveqres 6954
Description: Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
Assertion
Ref Expression
fveqres ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))

Proof of Theorem fveqres
StepHypRef Expression
1 fvres 6926 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 fvres 6926 . . . 4 (𝐴𝐵 → ((𝐺𝐵)‘𝐴) = (𝐺𝐴))
31, 2eqeq12d 2751 . . 3 (𝐴𝐵 → (((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴) ↔ (𝐹𝐴) = (𝐺𝐴)))
43biimprd 248 . 2 (𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
5 nfvres 6948 . . . 4 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
6 nfvres 6948 . . . 4 𝐴𝐵 → ((𝐺𝐵)‘𝐴) = ∅)
75, 6eqtr4d 2778 . . 3 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
87a1d 25 . 2 𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
94, 8pm2.61i 182 1 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  c0 4339  cres 5691  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-10 2139  ax-11 2155  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-nf 1781  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-xp 5695  df-dm 5699  df-res 5701  df-iota 6516  df-fv 6571
This theorem is referenced by:  fvresex  7983
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