Theorem aov0nbovbi 47110

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 47066	. 2 ⊢ (∅ ∉ 𝐶 → ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶))
2		df-aov 47036	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eleq1i 2835	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
4		df-ov 7451	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eleq1i 2835	. 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 2108   ∉ wnel 3052  ∅c0 4352  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448  '''cafv 47032   ((caov 47033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-aiota 47000  df-dfat 47034  df-afv 47035  df-aov 47036

This theorem is referenced by: (None)