Theorem aovpcov0 47653

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

Assertion

Ref	Expression
aovpcov0	⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)

Proof of Theorem aovpcov0

Step	Hyp	Ref	Expression
1		afvpcfv0 47609	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = V → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2		df-aov 47584	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eqeq1i 2744	. 2 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
4		df-ov 7359	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2744	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
6	1, 3, 5	3imtr4i 293	1 ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1547  Vcvv 3431  ∅c0 4261  ⟨cop 4561  ‘cfv 6485  (class class class)co 7356  '''cafv 47580   ((caov 47581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359  df-aiota 47548  df-dfat 47582  df-afv 47583  df-aov 47584

This theorem is referenced by: (None)