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Theorem ndmaovcom 46648
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7605. (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 5708 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
32eqcomi 2734 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
43eleq2i 2817 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
51, 4bitr3i 276 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaov 46626 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
75, 6sylnbi 329 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = V)
8 ancom 459 . . . 4 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
9 opelxp 5708 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐴𝑆))
103eleq2i 2817 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
118, 9, 103bitr2i 298 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12 ndmaov 46626 . . 3 (¬ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V)
1311, 12sylnbi 329 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐵𝐹𝐴)) = V)
147, 13eqtr4d 2768 1 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cop 4630   × cxp 5670  dom cdm 5672   ((caov 46561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 46528  df-dfat 46562  df-afv 46563  df-aov 46564
This theorem is referenced by: (None)
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