Theorem dfafv22 47885

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 47835	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		df-fv 6545	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
3	2	eqcomi 2778	. . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴)
4		ifeq1 4496	. . 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 2792	1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

Syntax hints:   = wceq 1567  ifcif 4492  𝒫 cpw 4567  ∪ cuni 4876   class class class wbr 5113  ran crn 5663  ℩cio 6491  ‘cfv 6537   defAt wdfat 47742  ''''cafv2 47834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-if 4493  df-fv 6545  df-afv2 47835

This theorem is referenced by:  dfatafv2eqfv  47887