Theorem dfatafv2ex 46493

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 46490	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		iotaex 6510	. 2 ⊢ (℩𝑥𝐴𝐹𝑥) ∈ V
3	1, 2	eqeltrdi 2835	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  ℩cio 6487   defAt wdfat 46396  ''''cafv2 46488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6489  df-afv2 46489

This theorem is referenced by:  dfatbrafv2b  46525  fnbrafv2b  46528  dfatdmfcoafv2  46534  dfatcolem  46535