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Theorem afv2ex 46468
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 46463 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iotaex 6507 . . . 4 (℩𝑥𝐴𝐹𝑥) ∈ V
32a1i 11 . . 3 (ran 𝐹𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4 uniexg 7724 . . . 4 (ran 𝐹𝑉 ran 𝐹 ∈ V)
54pwexd 5368 . . 3 (ran 𝐹𝑉 → 𝒫 ran 𝐹 ∈ V)
63, 5ifcld 4567 . 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 3466  ifcif 4521  𝒫 cpw 4595   cuni 4900   class class class wbr 5139  ran crn 5668  cio 6484   defAt wdfat 46370  ''''cafv2 46462
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-uni 4901  df-iota 6486  df-afv2 46463
This theorem is referenced by:  fexafv2ex  46474  fcdmvafv2v  46490
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