Theorem aov0nbovbi 47199

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 47155	. 2 ⊢ (∅ ∉ 𝐶 → ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶))
2		df-aov 47125	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eleq1i 2819	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
4		df-ov 7352	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eleq1i 2819	. 2 ⊢ ((𝐴𝐹𝐵) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶)
6	1, 3, 5	3bitr4g 314	1 ⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109   ∉ wnel 3029  ∅c0 4284  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349  '''cafv 47121   ((caov 47122

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 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-aiota 47089  df-dfat 47123  df-afv 47124  df-aov 47125

This theorem is referenced by: (None)