Theorem ndmaovg 47647

Description: The value of an operation outside its domain, analogous to ndmovg 7544. (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 5661	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆))
2		eleq2 2826	. . . . . 6 ⊢ ((𝑅 × 𝑆) = dom 𝐹 → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
3	2	eqcoms 2745	. . . . 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 47646	. 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 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   × cxp 5623  dom cdm 5625   ((caov 47581

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6449  df-fun 6495  df-fv 6501  df-aiota 47548  df-dfat 47582  df-afv 47583  df-aov 47584

This theorem is referenced by: (None)