Theorem fveqres 6928

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 6900	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		fvres 6900	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = (𝐺‘𝐴))
3	1, 2	eqeq12d 2752	. . 3 ⊢ (𝐴 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
4	3	biimprd 248	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
5		nfvres 6922	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6		nfvres 6922	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	eqtr4d 2774	. . 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 182	1 ⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4313   ↾ cres 5661  ‘cfv 6536

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-res 5671  df-iota 6489  df-fv 6544

This theorem is referenced by:  fvresex  7963