Theorem aovovn0oveq 41798

Description: If the operation's value at an argument is not the empty set, it equals the value of the alternative operation at this argument. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aovovn0oveq	⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Proof of Theorem aovovn0oveq

Step	Hyp	Ref	Expression
1		df-ov 6817	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 2996	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		afvfvn0fveq 41754	. . 3 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4		df-aov 41722	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	3, 4, 1	3eqtr4g 2819	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
6	2, 5	sylbi 207	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1632   ≠ wne 2932  ∅c0 4058  ⟨cop 4327  ‘cfv 6049  (class class class)co 6814  '''cafv 41718   ((caov 41719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-dfat 41720  df-afv 41721  df-aov 41722

This theorem is referenced by: (None)