Theorem elfv2ex 6934

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 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)

Proof of Theorem elfv2ex

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V))
2		fv2prc 6933	. . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅)
3	2	eleq2d 2819	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4		noel 4329	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
5	4	pm2.21i 119	. . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V)
6	3, 5	syl6bi 252	. 2 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V))
7	1, 6	pm2.61i 182	1 ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106  Vcvv 3474  ∅c0 4321  ‘cfv 6540

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-dm 5685  df-iota 6492  df-fv 6548

This theorem is referenced by: (None)