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Theorem afvvv 46311
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 46310 . . 3 𝐴 ∈ V → (𝐹'''𝐴) = V)
2 nvelim 46289 . . 3 ((𝐹'''𝐴) = V → ¬ (𝐹'''𝐴) ∈ 𝐵)
31, 2syl 17 . 2 𝐴 ∈ V → ¬ (𝐹'''𝐴) ∈ 𝐵)
43con4i 114 1 ((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2105  Vcvv 3473  '''cafv 46283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-aiota 46251  df-dfat 46285  df-afv 46286
This theorem is referenced by: (None)
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