Theorem fcdmvafv2v 46242

Description: If the codomain of a function is a set, the alternate function value is always also a set. (Contributed by AV, 4-Sep-2022.)

Assertion

Ref	Expression
fcdmvafv2v	⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V)

Proof of Theorem fcdmvafv2v

Step	Hyp	Ref	Expression
1		df-f 6546	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		ssexg 5322	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V)
3	2	ex 411	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V))
4	1, 3	simplbiim 503	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V))
5	4	imp 405	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V)
6		afv2ex 46220	. 2 ⊢ (ran 𝐹 ∈ V → (𝐹''''𝐶) ∈ V)
7	5, 6	syl 17	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104  Vcvv 3472   ⊆ wss 3947  ran crn 5676   Fn wfn 6537  ⟶wf 6538  ''''cafv2 46214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494  df-f 6546  df-afv2 46215

This theorem is referenced by: (None)