Theorem fveqres 6944

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 6916	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
2		fvres 6916	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = (𝐺‘𝐴))
3	1, 2	eqeq12d 2744	. . 3 ⊢ (𝐴 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
4	3	biimprd 247	. 2 ⊢ (𝐴 ∈ 𝐵 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴)))
5		nfvres 6938	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
6		nfvres 6938	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝐴) = ∅)
7	5, 6	eqtr4d 2771	. . 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 1534   ∈ wcel 2099  ∅c0 4323   ↾ cres 5680  ‘cfv 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-dm 5688  df-res 5690  df-iota 6500  df-fv 6556

This theorem is referenced by:  fvresex  7963