Theorem elfv2ex 6914

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 6913	. . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅)
3	2	eleq2d 2851	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4		noel 4293	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
5	4	pm2.21i 120	. . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V)
6	3, 5	biimtrdi 256	. 2 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V))
7	1, 6	pm2.61i 184	1 ⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-dm 5662  df-iota 6481  df-fv 6533

This theorem is referenced by:  2arwcat  50229