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Theorem dfafv22 43678
Description: Alternate definition of (𝐹''''𝐴) using (𝐹𝐴) directly. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
dfafv22 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)

Proof of Theorem dfafv22
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 43628 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 df-fv 6351 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2833 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 4453 . . 3 ((℩𝑥𝐴𝐹𝑥) = (𝐹𝐴) → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹))
53, 4ax-mp 5 . 2 if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
61, 5eqtri 2847 1 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  ifcif 4449  𝒫 cpw 4521   cuni 4824   class class class wbr 5052  ran crn 5543  cio 6300  cfv 6343   defAt wdfat 43535  ''''cafv2 43627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-un 3924  df-if 4450  df-fv 6351  df-afv2 43628
This theorem is referenced by:  dfatafv2eqfv  43680
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