Theorem afv2ex 47772

Description: The alternate function value is always a set if the range of the function is a set. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
afv2ex	⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Proof of Theorem afv2ex

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-afv2 47767	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iotaex 6493	. . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	2	a1i 11	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4		uniexg 7719	. . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
5	4	pwexd 5335	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V)
6	3, 5	ifcld 4526	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V)
7	1, 6	eqeltrid 2865	1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453  ifcif 4479  𝒫 cpw 4554  ∪ cuni 4864   class class class wbr 5099  ran crn 5646  ℩cio 6471   defAt wdfat 47674  ''''cafv2 47766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-uni 4865  df-iota 6473  df-afv2 47767

This theorem is referenced by:  fexafv2ex  47778  fcdmvafv2v  47794