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Theorem dfatafv2ex 45519
Description: The alternate function value at a class 𝐴 is always a set if the function/class 𝐹 is defined at 𝐴. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
dfatafv2ex (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

Proof of Theorem dfatafv2ex
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfatafv2iota 45516 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2 iotaex 6474 . 2 (℩𝑥𝐴𝐹𝑥) ∈ V
31, 2eqeltrdi 2846 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3448   class class class wbr 5110  cio 6451   defAt wdfat 45422  ''''cafv2 45514
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5268
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-uni 4871  df-iota 6453  df-afv2 45515
This theorem is referenced by:  dfatbrafv2b  45551  fnbrafv2b  45554  dfatdmfcoafv2  45560  dfatcolem  45561
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