Theorem afvvv 39874

Description: If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvvv	⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ V)

Proof of Theorem afvvv

Step	Hyp	Ref	Expression
1		afvprc 39873	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹'''𝐴) = V)
2		nvelim 39849	. . 3 ⊢ ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
3	1, 2	syl 17	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐹'''𝐴) ∈ 𝐵)
4	3	con4i 112	1 ⊢ ((𝐹'''𝐴) ∈ 𝐵 → 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  '''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: (None)