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Theorem dfatafv2ex 44159
 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 44156 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2 iotaex 6315 . 2 (℩𝑥𝐴𝐹𝑥) ∈ V
31, 2eqeltrdi 2860 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3409   class class class wbr 5032  ℩cio 6292   defAt wdfat 44062  ''''cafv2 44154 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-uni 4799  df-iota 6294  df-afv2 44155 This theorem is referenced by:  dfatbrafv2b  44191  fnbrafv2b  44194  dfatdmfcoafv2  44200  dfatcolem  44201
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