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Theorem ndmaovcom 43758
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7319. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1 dom 𝐹 = (𝑆 × 𝑆)
Assertion
Ref Expression
ndmaovcom (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Proof of Theorem ndmaovcom
StepHypRef Expression
1 opelxp 5559 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
32eqcomi 2810 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
43eleq2i 2884 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
51, 4bitr3i 280 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaov 43736 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
75, 6sylnbi 333 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = V)
8 ancom 464 . . . 4 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
9 opelxp 5559 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐴𝑆))
103eleq2i 2884 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
118, 9, 103bitr2i 302 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12 ndmaov 43736 . . 3 (¬ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V)
1311, 12sylnbi 333 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐵𝐹𝐴)) = V)
147, 13eqtr4d 2839 1 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534   × cxp 5521  dom cdm 5523   ((caov 43671
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-aiota 43639  df-dfat 43672  df-afv 43673  df-aov 43674
This theorem is referenced by: (None)
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