Theorem afv2ex 47205

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 47200	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iotaex 6486	. . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	2	a1i 11	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4		uniexg 7718	. . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
5	4	pwexd 5336	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V)
6	3, 5	ifcld 4537	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V)
7	1, 6	eqeltrid 2833	1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3450  ifcif 4490  𝒫 cpw 4565  ∪ cuni 4873   class class class wbr 5109  ran crn 5641  ℩cio 6464   defAt wdfat 47107  ''''cafv2 47199

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4874  df-iota 6466  df-afv2 47200

This theorem is referenced by:  fexafv2ex  47211  fcdmvafv2v  47227