Theorem aov0nbovbi 48091

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 48047	. 2 ⊢ (∅ ∉ 𝐶 → ((𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶))
2		df-aov 48017	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eleq1i 2853	. 2 ⊢ ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐹'''⟨𝐴, 𝐵⟩) ∈ 𝐶)
4		df-ov 7420	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eleq1i 2853	. 2 ⊢ ((𝐴𝐹𝐵) ∈ 𝐶 ↔ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶)
6	1, 3, 5	3bitr4g 317	1 ⊢ (∅ ∉ 𝐶 → ( ((𝐴𝐹𝐵)) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145   ∉ wnel 3063  ∅c0 4282  ⟨cop 4593  ‘cfv 6537  (class class class)co 7417  '''cafv 48013   ((caov 48014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-aiota 47981  df-dfat 48015  df-afv 48016  df-aov 48017

This theorem is used by: (None)