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Theorem elfv2ex 6490
Description: If a function value of a function value has a member, then 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 6489 . . . 4 𝐵 ∈ V → ((𝐹𝐵)‘𝐶) = ∅)
32eleq2d 2845 . . 3 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4 noel 4146 . . . 4 ¬ 𝐴 ∈ ∅
54pm2.21i 117 . . 3 (𝐴 ∈ ∅ → 𝐵 ∈ V)
63, 5syl6bi 245 . 2 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V))
71, 6pm2.61i 177 1 (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107  Vcvv 3398  c0 4141  cfv 6137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5027  ax-pow 5079
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-dm 5367  df-iota 6101  df-fv 6145
This theorem is referenced by: (None)
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