Theorem dfafv22 48468

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 48418	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		df-fv 6554	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
3	2	eqcomi 2772	. . 3 ⊢ (℩𝑥𝐴𝐹𝑥) = (𝐹‘𝐴)
4		ifeq1 4488	. . 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 2786	1 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), 𝒫 ∪ ran 𝐹)

This proof depends on syntax axioms:   = wceq 1569  ifcif 4484  𝒫 cpw 4559  ∪ cuni 4869   class class class wbr 5105  ran crn 5655  ℩cio 6498  ‘cfv 6546   defAt wdfat 48326  ''''cafv2 48417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-un 4153  df-if 4485  df-fv 6554  df-afv2 48418

This theorem is used by:  dfatafv2eqfv  48470