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Theorem elfv2ex 6948
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 6947 . . . 4 𝐵 ∈ V → ((𝐹𝐵)‘𝐶) = ∅)
32eleq2d 2815 . . 3 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4 noel 4334 . . . 4 ¬ 𝐴 ∈ ∅
54pm2.21i 119 . . 3 (𝐴 ∈ ∅ → 𝐵 ∈ V)
63, 5biimtrdi 252 . 2 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V))
71, 6pm2.61i 182 1 (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2098  Vcvv 3473  c0 4326  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561
This theorem is referenced by: (None)
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