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Theorem aovnuoveq 44164
 Description: The alternative value of the operation on an ordered pair equals the operation's value at this ordered pair. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovnuoveq ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))

Proof of Theorem aovnuoveq
StepHypRef Expression
1 df-aov 44094 . . 3 ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
21neeq1i 3015 . 2 ( ((𝐴𝐹𝐵)) ≠ V ↔ (𝐹'''⟨𝐴, 𝐵⟩) ≠ V)
3 afvnufveq 44120 . . 3 ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → (𝐹'''⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐴, 𝐵⟩))
4 df-ov 7159 . . 3 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
53, 1, 43eqtr4g 2818 . 2 ((𝐹'''⟨𝐴, 𝐵⟩) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
62, 5sylbi 220 1 ( ((𝐴𝐹𝐵)) ≠ V → ((𝐴𝐹𝐵)) = (𝐴𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ≠ wne 2951  Vcvv 3409  ⟨cop 4531  ‘cfv 6340  (class class class)co 7156  '''cafv 44090   ((caov 44091 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 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302 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 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-aiota 44057  df-dfat 44092  df-afv 44093  df-aov 44094 This theorem is referenced by: (None)
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