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Theorem afv2ex 43279
 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 43274 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iotaex 6333 . . . 4 (℩𝑥𝐴𝐹𝑥) ∈ V
32a1i 11 . . 3 (ran 𝐹𝑉 → (℩𝑥𝐴𝐹𝑥) ∈ V)
4 uniexg 7457 . . . 4 (ran 𝐹𝑉 ran 𝐹 ∈ V)
54pwexd 5277 . . 3 (ran 𝐹𝑉 → 𝒫 ran 𝐹 ∈ V)
63, 5ifcld 4515 . 2 (ran 𝐹𝑉 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) ∈ V)
71, 6eqeltrid 2922 1 (ran 𝐹𝑉 → (𝐹''''𝐴) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3500  ifcif 4470  𝒫 cpw 4542  ∪ cuni 4837   class class class wbr 5063  ran crn 5555  ℩cio 6310   defAt wdfat 43181  ''''cafv2 43273 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-uni 4838  df-iota 6312  df-afv2 43274 This theorem is referenced by:  fexafv2ex  43285  frnvafv2v  43301
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