Theorem dfatafv2ex 48227

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 48224	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		iotaex 6507	. 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	1, 2	eqeltrdi 2869	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  ℩cio 6485   defAt wdfat 48130  ''''cafv2 48222

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6487  df-afv2 48223

This theorem is used by:  dfatbrafv2b  48259  fnbrafv2b  48262  dfatdmfcoafv2  48268  dfatcolem  48269