Theorem elfv2ex 6713

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 6712	. . . 4 ⊢ (¬ 𝐵 ∈ V → ((𝐹‘𝐵)‘𝐶) = ∅)
3	2	eleq2d 2900	. . 3 ⊢ (¬ 𝐵 ∈ V → (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) ↔ 𝐴 ∈ ∅))
4		noel 4298	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
5	4	pm2.21i 119	. . 3 ⊢ (𝐴 ∈ ∅ → 𝐵 ∈ V)
6	3, 5	syl6bi 255	. 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 2114  Vcvv 3496  ∅c0 4293  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-dm 5567  df-iota 6316  df-fv 6365

This theorem is referenced by: (None)