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Theorem elfv2ex 6196
 Description: If a function value of a function value has a member, the first argument is a set. (Contributed by AV, 8-Apr-2021.)
Assertion
Ref Expression
elfv2ex (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)

Proof of Theorem elfv2ex
StepHypRef Expression
1 ax-1 6 . 2 (𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V))
2 fv2prc 6195 . . . 4 𝐵 ∈ V → ((𝐹𝐵)‘𝐶) = ∅)
32eleq2d 2684 . . 3 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4 noel 3901 . . . 4 ¬ 𝐴 ∈ ∅
54pm2.21i 116 . . 3 (𝐴 ∈ ∅ → 𝐵 ∈ V)
63, 5syl6bi 243 . 2 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V))
71, 6pm2.61i 176 1 (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1987  Vcvv 3190  ∅c0 3897  ‘cfv 5857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759  ax-pow 4813 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-dm 5094  df-iota 5820  df-fv 5865 This theorem is referenced by: (None)
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