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Theorem fveqres 6949
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 6921 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 fvres 6921 . . . 4 (𝐴𝐵 → ((𝐺𝐵)‘𝐴) = (𝐺𝐴))
31, 2eqeq12d 2744 . . 3 (𝐴𝐵 → (((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴) ↔ (𝐹𝐴) = (𝐺𝐴)))
43biimprd 247 . 2 (𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
5 nfvres 6943 . . . 4 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
6 nfvres 6943 . . . 4 𝐴𝐵 → ((𝐺𝐵)‘𝐴) = ∅)
75, 6eqtr4d 2771 . . 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 1533  wcel 2098  c0 4326  cres 5684  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-11 2146  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-xp 5688  df-dm 5692  df-res 5694  df-iota 6505  df-fv 6561
This theorem is referenced by:  fvresex  7969
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