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Theorem ndmaovg 43737
Description: The value of an operation outside its domain, analogous to ndmovg 7315. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
ndmaovg ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)

Proof of Theorem ndmaovg
StepHypRef Expression
1 opelxp 5559 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ (𝐴𝑅𝐵𝑆))
2 eleq2 2881 . . . . . 6 ((𝑅 × 𝑆) = dom 𝐹 → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
32eqcoms 2809 . . . . 5 (dom 𝐹 = (𝑅 × 𝑆) → (⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
41, 3bitr3id 288 . . . 4 (dom 𝐹 = (𝑅 × 𝑆) → ((𝐴𝑅𝐵𝑆) ↔ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
54notbid 321 . . 3 (dom 𝐹 = (𝑅 × 𝑆) → (¬ (𝐴𝑅𝐵𝑆) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹))
65biimpa 480 . 2 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹)
7 ndmaov 43736 . 2 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)
86, 7syl 17 1 ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴𝑅𝐵𝑆)) → ((𝐴𝐹𝐵)) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  cop 4534   × cxp 5521  dom cdm 5523   ((caov 43671
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-aiota 43639  df-dfat 43672  df-afv 43673  df-aov 43674
This theorem is referenced by: (None)
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