Theorem dfatafv2ex 47225

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 47222	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		iotaex 6534	. 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	1, 2	eqeltrdi 2849	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  ℩cio 6512   defAt wdfat 47128  ''''cafv2 47220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514  df-afv2 47221

This theorem is referenced by:  dfatbrafv2b  47257  fnbrafv2b  47260  dfatdmfcoafv2  47266  dfatcolem  47267