Theorem fveqres 6926

Description: Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
fveqres	⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))

Proof of Theorem fveqres

Step	Hyp	Ref	Expression
1		fvres 6901	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		fvres 6901	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = (𝐺‘𝐴))
3	1, 2	eqeq12d 2785	. . 3 ⊢ (𝐴 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
4	3	biimprd 251	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
5		nfvres 6920	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6		nfvres 6920	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	eqtr4d 2807	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))
8	7	a1d 26	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
9	4, 8	pm2.61i 184	1 ⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  ∅c0 4294   ↾ cres 5664  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-dm 5672  df-res 5674  df-iota 6493  df-fv 6545

This theorem is referenced by:  fvresex  7957