Theorem ndmaovg 47169

Description: The value of an operation outside its domain, analogous to ndmovg 7613. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
ndmaovg	⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)

Proof of Theorem ndmaovg

Step	Hyp	Ref	Expression
1		opelxp 5719	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
2		eleq2 2829	. . . . . 6 ⊢ ((𝑅 × 𝑆) = dom 𝐹 → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
3	2	eqcoms 2744	. . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
4	1, 3	bitr3id 285	. . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
5	4	notbid 318	. . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
6	5	biimpa 476	. 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
7		ndmaov 47168	. 2 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
8	6, 7	syl 17	1 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ((𝐴𝐹𝐵)) = V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3479  ⟨cop 4630   × cxp 5681  dom cdm 5683   ((caov 47103

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 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-res 5695  df-iota 6512  df-fun 6561  df-fv 6567  df-aiota 47070  df-dfat 47104  df-afv 47105  df-aov 47106

This theorem is referenced by: (None)