Theorem dfatafv2ex 47243

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 47240	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		iotaex 6457	. 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	1, 2	eqeltrdi 2839	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436   class class class wbr 5091  ℩cio 6435   defAt wdfat 47146  ''''cafv2 47238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-uni 4860  df-iota 6437  df-afv2 47239

This theorem is referenced by:  dfatbrafv2b  47275  fnbrafv2b  47278  dfatdmfcoafv2  47284  dfatcolem  47285