Theorem aovpcov0 47352

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 47308	. 2 ⊢ ((𝐹'''⟨𝐴, 𝐵⟩) = V → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2		df-aov 47283	. . 3 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
3	2	eqeq1i 2738	. 2 ⊢ ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
4		df-ov 7358	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2738	. 2 ⊢ ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
6	1, 3, 5	3imtr4i 292	1 ⊢ ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3437  ∅c0 4282  ⟨cop 4583  ‘cfv 6489  (class class class)co 7355  '''cafv 47279   ((caov 47280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-aiota 47247  df-dfat 47281  df-afv 47282  df-aov 47283

This theorem is referenced by: (None)