Theorem aov0ov0 44929

Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
aov0ov0	⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)

Proof of Theorem aov0ov0

Step	Hyp	Ref	Expression
1		afv0fv0 44885	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2		df-aov 44857	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eqeq1i 2741	. 2 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
4		df-ov 7310	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2741	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
6	1, 3, 5	3imtr4i 292	1 ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1539  ∅c0 4262  ⟨cop 4571  ‘cfv 6458  (class class class)co 7307  '''cafv 44853   ((caov 44854

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-res 5612  df-iota 6410  df-fun 6460  df-fv 6466  df-ov 7310  df-aiota 44821  df-dfat 44855  df-afv 44856  df-aov 44857

This theorem is referenced by: (None)