Theorem afv2ex 47806

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 47801	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iotaex 6501	. . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	2	a1i 11	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4		uniexg 7727	. . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
5	4	pwexd 5341	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V)
6	3, 5	ifcld 4530	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V)
7	1, 6	eqeltrid 2869	1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457  ifcif 4483  𝒫 cpw 4558  ∪ cuni 4868   class class class wbr 5105  ran crn 5653  ℩cio 6479   defAt wdfat 47708  ''''cafv2 47800

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-sep 5251  ax-nul 5261  ax-pow 5327  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-uni 4869  df-iota 6481  df-afv2 47801

This theorem is referenced by:  fexafv2ex  47812  fcdmvafv2v  47828