Theorem ndmaovcom 47155

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	⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Proof of Theorem ndmaovcom

Step	Hyp	Ref	Expression
1		opelxp 5725	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
2		ndmaov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
3	2	eqcomi 2744	. . . . 5 ⊢ (𝑆 × 𝑆) = dom 𝐹
4	3	eleq2i 2831	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
5	1, 4	bitr3i 277	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
6		ndmaov 47133	. . 3 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
7	5, 6	sylnbi 330	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = V)
8		ancom 460	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
9		opelxp 5725	. . . 4 ⊢ (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆))
10	3	eleq2i 2831	. . . 4 ⊢ (⟨𝐵, 𝐴⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
11	8, 9, 10	3bitr2i 299	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ↔ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹)
12		ndmaov 47133	. . 3 ⊢ (¬ ⟨𝐵, 𝐴⟩ ∈ dom 𝐹 → ((𝐵𝐹𝐴)) = V)
13	11, 12	sylnbi 330	. 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐵𝐹𝐴)) = V)
14	7, 13	eqtr4d 2778	1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵)) = ((𝐵𝐹𝐴)) )

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637   × cxp 5687  dom cdm 5689   ((caov 47068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070  df-aov 47071

This theorem is referenced by: (None)