Theorem fveqres 6389

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 6366	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		fvres 6366	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = (𝐺‘𝐴))
3	1, 2	eqeq12d 2773	. . 3 ⊢ (𝐴 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
4	3	biimprd 238	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
5		nfvres 6383	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6		nfvres 6383	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	eqtr4d 2795	. . 3 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))
8	7	a1d 25	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
9	4, 8	pm2.61i 176	1 ⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1630   ∈ wcel 2137  ∅c0 4056   ↾ cres 5266  ‘cfv 6047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-xp 5270  df-dm 5274  df-res 5276  df-iota 6010  df-fv 6055

This theorem is referenced by:  fvresex  7302