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Theorem dfafv22 45565
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 45515 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 df-fv 6509 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2746 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 4495 . . 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 2765 1 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  ifcif 4491  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  ran crn 5639  cio 6451  cfv 6501   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-un 3920  df-if 4492  df-fv 6509  df-afv2 45515
This theorem is referenced by:  dfatafv2eqfv  45567
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