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Theorem afv2ex 46764
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 𝐹𝑉 → (𝐹''''𝐴) ∈ V)

Proof of Theorem afv2ex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 46759 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iotaex 6526 . . . 4 (℩𝑥𝐴𝐹𝑥) ∈ V
32a1i 11 . . 3 (ran 𝐹𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4 uniexg 7750 . . . 4 (ran 𝐹𝑉 ran 𝐹 ∈ V)
54pwexd 5382 . . 3 (ran 𝐹𝑉 → 𝒫 ran 𝐹 ∈ V)
63, 5ifcld 4578 . 2 (ran 𝐹𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) ∈ V)
71, 6eqeltrid 2829 1 (ran 𝐹𝑉 → (𝐹''''𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3461  ifcif 4532  𝒫 cpw 4606   cuni 4912   class class class wbr 5152  ran crn 5682  cio 6503   defAt wdfat 46666  ''''cafv2 46758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-uni 4913  df-iota 6505  df-afv2 46759
This theorem is referenced by:  fexafv2ex  46770  fcdmvafv2v  46786
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