Theorem aov0nbovbi 47659

Description: The operation's value on an ordered pair is an element of a set if and only if the alternative value of the operation on this ordered pair is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aov0nbovbi	⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))

Proof of Theorem aov0nbovbi

Step	Hyp	Ref	Expression
1		afv0nbfvbi 47615	. 2 ⊢ (∅ ∉ 𝐶 → ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶))
2		df-aov 47585	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eleq1i 2831	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
4		df-ov 7366	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eleq1i 2831	. 2 ⊢ ((𝐴𝐹𝐵) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶)
6	1, 3, 5	3bitr4g 315	1 ⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2119   ∉ wnel 3039  ∅c0 4268  ⟨cop 4568  ‘cfv 6492  (class class class)co 7363  '''cafv 47581   ((caov 47582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-aiota 47549  df-dfat 47583  df-afv 47584  df-aov 47585

This theorem is referenced by: (None)