Theorem elfv2ex 6877

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 6876	. . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅)
3	2	eleq2d 2822	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4		noel 4290	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
5	4	pm2.21i 119	. . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V)
6	3, 5	biimtrdi 253	. 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 2113  Vcvv 3440  ∅c0 4285  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-dm 5634  df-iota 6448  df-fv 6500

This theorem is referenced by:  2arwcat  49841