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Theorem dfatafv2iota 43557
Description: If a function is defined at a class 𝐴 the alternate function value at 𝐴 is the unique value assigned to 𝐴 by the function (analogously to (𝐹𝐴)). (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
dfatafv2iota (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfatafv2iota
StepHypRef Expression
1 df-afv2 43556 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iftrue 4449 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
31, 2syl5eq 2867 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  ifcif 4443  𝒫 cpw 4515   cuni 4814   class class class wbr 5042  ran crn 5532  cio 6288   defAt wdfat 43463  ''''cafv2 43555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-if 4444  df-afv2 43556
This theorem is referenced by:  dfatafv2ex  43560  funressndmafv2rn  43570  afv2eu  43585  afv2res  43586  tz6.12-afv2  43587  dfafv23  43600  rlimdmafv2  43605
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