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Theorem elfv2ex 6684
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 6683 . . . 4 𝐵 ∈ V → ((𝐹𝐵)‘𝐶) = ∅)
32eleq2d 2897 . . 3 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4 noel 4270 . . . 4 ¬ 𝐴 ∈ ∅
54pm2.21i 119 . . 3 (𝐴 ∈ ∅ → 𝐵 ∈ V)
63, 5syl6bi 256 . 2 𝐵 ∈ V → (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V))
71, 6pm2.61i 185 1 (𝐴 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2115  Vcvv 3471  c0 4266  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183  ax-pow 5239
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-dm 5538  df-iota 6287  df-fv 6336
This theorem is referenced by: (None)
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