MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fveqres Structured version   Visualization version   GIF version

Theorem fveqres 6953
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 6925 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
2 fvres 6925 . . . 4 (𝐴𝐵 → ((𝐺𝐵)‘𝐴) = (𝐺𝐴))
31, 2eqeq12d 2753 . . 3 (𝐴𝐵 → (((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴) ↔ (𝐹𝐴) = (𝐺𝐴)))
43biimprd 248 . 2 (𝐴𝐵 → ((𝐹𝐴) = (𝐺𝐴) → ((𝐹𝐵)‘𝐴) = ((𝐺𝐵)‘𝐴)))
5 nfvres 6947 . . . 4 𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)
6 nfvres 6947 . . . 4 𝐴𝐵 → ((𝐺𝐵)‘𝐴) = ∅)
75, 6eqtr4d 2780 . . 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 1540  wcel 2108  c0 4333  cres 5687  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-10 2141  ax-11 2157  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-nf 1784  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-xp 5691  df-dm 5695  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  fvresex  7984
  Copyright terms: Public domain W3C validator