Theorem ndmaovg 45882

Description: The value of an operation outside its domain, analogous to ndmovg 7589. (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 5712	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
2		eleq2 2822	. . . . . 6 ⊢ ((𝑅 × 𝑆) = dom 𝐹 → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
3	2	eqcoms 2740	. . . . 5 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
4	1, 3	bitr3id 284	. . . 4 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
5	4	notbid 317	. . 3 ⊢ (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
6	5	biimpa 477	. 2 ⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
7		ndmaov 45881	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4634   × cxp 5674  dom cdm 5676   ((caov 45816

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 45783  df-dfat 45817  df-afv 45818  df-aov 45819

This theorem is referenced by: (None)