Theorem dfafv22 47276

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 47226	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		df-fv 6568	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
3	2	eqcomi 2745	. . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴)
4		ifeq1 4528	. . 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 2764	1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

Syntax hints:   = wceq 1539  ifcif 4524  𝒫 cpw 4599  ∪ cuni 4906   class class class wbr 5142  ran crn 5685  ℩cio 6511  ‘cfv 6560   defAt wdfat 47133  ''''cafv2 47225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-un 3955  df-if 4525  df-fv 6568  df-afv2 47226

This theorem is referenced by:  dfatafv2eqfv  47278