Theorem afv2ex 48499

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 48494	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iotaex 6536	. . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	2	a1i 11	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4		uniexg 7784	. . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
5	4	pwexd 5346	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V)
6	3, 5	ifcld 4532	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V)
7	1, 6	eqeltrid 2868	1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454  ifcif 4485  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5106  ran crn 5657  ℩cio 6513   defAt wdfat 48402  ''''cafv2 48493

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6515  df-afv2 48494

This theorem is used by:  fexafv2ex  48505  fcdmvafv2v  48521