Theorem dfafv22 47247

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 47197	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		df-fv 6490	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
3	2	eqcomi 2738	. . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴)
4		ifeq1 4480	. . 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 2752	1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

Syntax hints:   = wceq 1540  ifcif 4476  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  ran crn 5620  ℩cio 6436  ‘cfv 6482   defAt wdfat 47104  ''''cafv2 47196

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-un 3908  df-if 4477  df-fv 6490  df-afv2 47197

This theorem is referenced by:  dfatafv2eqfv  47249