Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvvdm Structured version   Visualization version   GIF version

Theorem afvvdm 39870
 Description: If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvvdm ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)

Proof of Theorem afvvdm
StepHypRef Expression
1 ndmafv 39869 . . 3 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
2 nvelim 39849 . . 3 ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
31, 2syl 17 . 2 𝐴 ∈ dom 𝐹 → ¬ (𝐹'''𝐴) ∈ 𝐵)
43con4i 112 1 ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  dom cdm 5038  '''cafv 39843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037  df-fv 5812  df-dfat 39845  df-afv 39846 This theorem is referenced by:  aovvdm  39914  aovrcl  39918  aoprssdm  39931
 Copyright terms: Public domain W3C validator