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Theorem fveqres 6706
 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 6683 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 fvres 6683 . . . 4 (𝐴𝐵 → ((𝐺𝐵)‘𝐴) = (𝐺𝐴))
31, 2eqeq12d 2775 . . 3 (𝐴𝐵 → (((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴) ↔ (𝐹𝐴) = (𝐺𝐴)))
43biimprd 251 . 2 (𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
5 nfvres 6700 . . . 4 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
6 nfvres 6700 . . . 4 𝐴𝐵 → ((𝐺𝐵)‘𝐴) = ∅)
75, 6eqtr4d 2797 . . 3 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
87a1d 25 . 2 𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
94, 8pm2.61i 185 1 ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2112  ∅c0 4228   ↾ cres 5531  ‘cfv 6341 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-xp 5535  df-dm 5539  df-res 5541  df-iota 6300  df-fv 6349 This theorem is referenced by:  fvresex  7672
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