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Theorem afvvv 39672
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) ∈ 𝐵𝐴 ∈ V)

Proof of Theorem afvvv
StepHypRef Expression
1 afvprc 39671 . . 3 𝐴 ∈ V → (𝐹'''𝐴) = V)
2 nvelim 39646 . . 3 ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
31, 2syl 17 . 2 𝐴 ∈ V → ¬ (𝐹'''𝐴) ∈ 𝐵)
43con4i 111 1 ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  '''cafv 39640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-un 3544  df-if 4036  df-fv 5798  df-dfat 39642  df-afv 39643
This theorem is referenced by: (None)
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