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Theorem dfatafv2iota 45516
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 45515 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iftrue 4497 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
31, 2eqtrid 2789 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  ifcif 4491  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  ran crn 5639  cio 6451   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-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-if 4492  df-afv2 45515
This theorem is referenced by:  dfatafv2ex  45519  funressndmafv2rn  45529  afv2eu  45544  afv2res  45545  tz6.12-afv2  45546  dfafv23  45559  rlimdmafv2  45564
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