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Theorem dfatafv2iota 47160
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 47159 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iftrue 4537 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
31, 2eqtrid 2787 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  ifcif 4531  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  ran crn 5690  cio 6514   defAt wdfat 47066  ''''cafv2 47158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-if 4532  df-afv2 47159
This theorem is referenced by:  dfatafv2ex  47163  funressndmafv2rn  47173  afv2eu  47188  afv2res  47189  tz6.12-afv2  47190  dfafv23  47203  rlimdmafv2  47208
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