Theorem afv2ex 44286

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 44281	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iotaex 6329	. . . 4 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	2	a1i 11	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4		uniexg 7496	. . . 4 ⊢ (ran 𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
5	4	pwexd 5256	. . 3 ⊢ (ran 𝐹 ∈ 𝑉 → 𝒫 ∪ ran 𝐹 ∈ V)
6	3, 5	ifcld 4470	. 2 ⊢ (ran 𝐹 ∈ 𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) ∈ V)
7	1, 6	eqeltrid 2838	1 ⊢ (ran 𝐹 ∈ 𝑉 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3400  ifcif 4424  𝒫 cpw 4498  ∪ cuni 4806   class class class wbr 5040  ran crn 5536  ℩cio 6305   defAt wdfat 44188  ''''cafv2 44280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-uni 4807  df-iota 6307  df-afv2 44281

This theorem is referenced by:  fexafv2ex  44292  frnvafv2v  44308