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Theorem ndmaovcom 45407
Description: Any operation is commutative outside its domain, analogous to ndmovcom 7538. (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 5668 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴𝑆𝐵𝑆))
2 ndmaov.1 . . . . . 6 dom 𝐹 = (𝑆 × 𝑆)
32eqcomi 2745 . . . . 5 (𝑆 × 𝑆) = dom 𝐹
43eleq2i 2829 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
51, 4bitr3i 276 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6 ndmaov 45385 . . 3 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
75, 6sylnbi 329 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = V)
8 ancom 461 . . . 4 ((𝐴𝑆𝐵𝑆) ↔ (𝐵𝑆𝐴𝑆))
9 opelxp 5668 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵𝑆𝐴𝑆))
103eleq2i 2829 . . . 4 (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
118, 9, 103bitr2i 298 . . 3 ((𝐴𝑆𝐵𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12 ndmaov 45385 . . 3 (¬ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V)
1311, 12sylnbi 329 . 2 (¬ (𝐴𝑆𝐵𝑆) → ((𝐵𝐹𝐴)) = V)
147, 13eqtr4d 2779 1 (¬ (𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3444  cop 4591   × cxp 5630  dom cdm 5632   ((caov 45320
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6446  df-fun 6496  df-fv 6502  df-aiota 45287  df-dfat 45321  df-afv 45322  df-aov 45323
This theorem is referenced by: (None)
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