Theorem aovovn0oveq 47224

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 7349	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2	1	neeq1i 2992	. 2 ⊢ ((𝐴𝐹𝐵) ≠ ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅)
3		afvfvn0fveq 47180	. . 3 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4		df-aov 47151	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
5	3, 4, 1	3eqtr4g 2791	. 2 ⊢ ((𝐹‘⟨𝐴, 𝐵⟩) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
6	2, 5	sylbi 217	1 ⊢ ((𝐴𝐹𝐵) ≠ ∅ → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Syntax hints:   → wi 4   = wceq 1541   ≠ wne 2928  ∅c0 4283  ⟨cop 4582  ‘cfv 6481  (class class class)co 7346  '''cafv 47147   ((caov 47148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-res 5628  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-aiota 47115  df-dfat 47149  df-afv 47150  df-aov 47151

This theorem is referenced by: (None)