Theorem ndmafv 41541

Description: The value of a class outside its domain is the universe, compare with ndmfv 6256. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
ndmafv	⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Proof of Theorem ndmafv

Step	Hyp	Ref	Expression
1		df-dfat 41517	. . . 4 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2	1	simplbi 475	. . 3 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹)
3	2	con3i 150	. 2 ⊢ (¬ 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴)
4		afvnfundmuv 41540	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
5	3, 4	syl 17	1 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231  {csn 4210  dom cdm 5143   ↾ cres 5145  Fun wfun 5920   defAt wdfat 41514  '''cafv 41515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-un 3612  df-if 4120  df-fv 5934  df-dfat 41517  df-afv 41518

This theorem is referenced by:  afvvdm  41542  afvprc  41545  afvco2  41577  ndmaov  41584  aovprc  41589