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Theorem afv2ex 44706
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 44701 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iotaex 6413 . . . 4 (℩𝑥𝐴𝐹𝑥) ∈ V
32a1i 11 . . 3 (ran 𝐹𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4 uniexg 7593 . . . 4 (ran 𝐹𝑉 ran 𝐹 ∈ V)
54pwexd 5302 . . 3 (ran 𝐹𝑉 → 𝒫 ran 𝐹 ∈ V)
63, 5ifcld 4505 . 2 (ran 𝐹𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) ∈ V)
71, 6eqeltrid 2843 1 (ran 𝐹𝑉 → (𝐹''''𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3432  ifcif 4459  𝒫 cpw 4533   cuni 4839   class class class wbr 5074  ran crn 5590  cio 6389   defAt wdfat 44608  ''''cafv2 44700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391  df-afv2 44701
This theorem is referenced by:  fexafv2ex  44712  frnvafv2v  44728
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