Theorem aov0ov0 45887

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 45843	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = ∅ → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2		df-aov 45815	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eqeq1i 2737	. 2 ⊢ ( ((𝐴𝐹𝐵)) = ∅ ↔ (𝐹'''⟨𝐴, 𝐵⟩) = ∅)
4		df-ov 7408	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2737	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
6	1, 3, 5	3imtr4i 291	1 ⊢ ( ((𝐴𝐹𝐵)) = ∅ → (𝐴𝐹𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1541  ∅c0 4321  ⟨cop 4633  ‘cfv 6540  (class class class)co 7405  '''cafv 45811   ((caov 45812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-aiota 45779  df-dfat 45813  df-afv 45814  df-aov 45815

This theorem is referenced by: (None)