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Theorem dfafv22 42159
 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, (𝐹𝐴), 𝒫 ran 𝐹)

Proof of Theorem dfafv22
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 42109 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 df-fv 6135 . . . 4 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
32eqcomi 2834 . . 3 (℩𝑥𝐴𝐹𝑥) = (𝐹𝐴)
4 ifeq1 4312 . . 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 2849 1 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1656  ifcif 4308  𝒫 cpw 4380  ∪ cuni 4660   class class class wbr 4875  ran crn 5347  ℩cio 6088  ‘cfv 6127   defAt wdfat 42016  ''''cafv2 42108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-un 3803  df-if 4309  df-fv 6135  df-afv2 42109 This theorem is referenced by:  dfatafv2eqfv  42161
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