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Theorem aovpcov0 45496
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 45452 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) = V → (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
2 df-aov 45427 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
32eqeq1i 2742 . 2 ( ((𝐴𝐹𝐵)) = V ↔ (𝐹'''⟨𝐴, 𝐵⟩) = V)
4 df-ov 7365 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
54eqeq1i 2742 . 2 ((𝐴𝐹𝐵) = ∅ ↔ (𝐹‘⟨𝐴, 𝐵⟩) = ∅)
61, 3, 53imtr4i 292 1 ( ((𝐴𝐹𝐵)) = V → (𝐴𝐹𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3448  c0 4287  cop 4597  cfv 6501  (class class class)co 7362  '''cafv 45423   ((caov 45424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-aiota 45391  df-dfat 45425  df-afv 45426  df-aov 45427
This theorem is referenced by: (None)
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