Theorem dfatafv2iota 47222

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 (𝐹‘𝐴)). (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

Step	Hyp	Ref	Expression
1		df-afv2 47221	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iftrue 4531	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
3	1, 2	eqtrid 2789	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))

Syntax hints:   → wi 4   = wceq 1540  ifcif 4525  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143  ran crn 5686  ℩cio 6512   defAt wdfat 47128  ''''cafv2 47220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526  df-afv2 47221

This theorem is referenced by:  dfatafv2ex  47225  funressndmafv2rn  47235  afv2eu  47250  afv2res  47251  tz6.12-afv2  47252  dfafv23  47265  rlimdmafv2  47270