Theorem dfatafv2ex 45921

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.

Step	Hyp	Ref	Expression
1		dfatafv2iota 45918	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		iotaex 6517	. 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	1, 2	eqeltrdi 2842	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  ℩cio 6494   defAt wdfat 45824  ''''cafv2 45916

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 2704  ax-nul 5307

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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496  df-afv2 45917

This theorem is referenced by:  dfatbrafv2b  45953  fnbrafv2b  45956  dfatdmfcoafv2  45962  dfatcolem  45963