Theorem ndmaov 48058

Description: The value of an operation outside its domain, analogous to ndmafv 48015. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
ndmaov	⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

Proof of Theorem ndmaov

Step	Hyp	Ref	Expression
1		df-aov 47996	. 2 ⊢ ((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)
2		ndmafv 48015	. 2 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → (𝐹'''⟨𝐴, 𝐵⟩) = V)
3	1, 2	eqtrid 2809	1 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝐹 → ((𝐴𝐹𝐵)) = V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ⟨cop 4593  dom cdm 5659  '''cafv 47992   ((caov 47993

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6493  df-fun 6539  df-fv 6545  df-aiota 47960  df-dfat 47994  df-afv 47995  df-aov 47996

This theorem is used by:  ndmaovg  48059  ndmaovcl  48078  ndmaovcom  48080  ndmaovass  48081  ndmaovdistr  48082