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Theorem ndmaovcom 47217
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7620. (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 5721 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
32eqcomi 2746 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
43eleq2i 2833 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
51, 4bitr3i 277 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaov 47195 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
75, 6sylnbi 330 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = V)
8 ancom 460 . . . 4 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
9 opelxp 5721 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐴𝑆))
103eleq2i 2833 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
118, 9, 103bitr2i 299 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12 ndmaov 47195 . . 3 (¬ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V)
1311, 12sylnbi 330 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐵𝐹𝐴)) = V)
147, 13eqtr4d 2780 1 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   × cxp 5683  dom cdm 5685   ((caov 47130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132  df-aov 47133
This theorem is referenced by: (None)
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