Theorem dfafv22 46265

Description: Alternate definition of (𝐹''''𝐴) using (𝐹‘𝐴) directly. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
dfafv22	⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

Proof of Theorem dfafv22

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-afv2 46215	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		df-fv 6550	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
3	2	eqcomi 2739	. . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴)
4		ifeq1 4531	. . 3 ⊢ ((℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴) → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹))
5	3, 4	ax-mp 5	. 2 ⊢ if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)
6	1, 5	eqtri 2758	1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

Syntax hints:   = wceq 1539  ifcif 4527  𝒫 cpw 4601  ∪ cuni 4907   class class class wbr 5147  ran crn 5676  ℩cio 6492  ‘cfv 6542   defAt wdfat 46122  ''''cafv2 46214

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-un 3952  df-if 4528  df-fv 6550  df-afv2 46215

This theorem is referenced by:  dfatafv2eqfv  46267