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Theorem aovpcov0 43744
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
StepHypRef Expression
1 afvpcfv0 43700 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) = V → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2 df-aov 43675 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
32eqeq1i 2803 . 2 ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
4 df-ov 7138 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
54eqeq1i 2803 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
61, 3, 53imtr4i 295 1 ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  Vcvv 3441  c0 4243  cop 4531  cfv 6324  (class class class)co 7135  '''cafv 43671   ((caov 43672
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-aiota 43640  df-dfat 43673  df-afv 43674  df-aov 43675
This theorem is referenced by: (None)
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