Theorem fcdmvafv2v 48109

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 6541	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
2		ssexg 5288	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V)
3	2	ex 418	. . . 4 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V))
4	1, 3	simplbiim 514	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐵 ∈ 𝑉 → ran 𝐹 ∈ V))
5	4	imp 412	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → ran 𝐹 ∈ V)
6		afv2ex 48087	. 2 ⊢ (ran 𝐹 ∈ V → (𝐹''''𝐶) ∈ V)
7	5, 6	syl 18	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ∈ 𝑉) → (𝐹''''𝐶) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ran crn 5660   Fn wfn 6532  ⟶wf 6533  ''''cafv2 48081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-uni 4871  df-iota 6493  df-f 6541  df-afv2 48082

This theorem is used by: (None)