Theorem dfatafv2iota 47456

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 47455	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iftrue 4485	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
3	1, 2	eqtrid 2783	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))

Syntax hints:   → wi 4   = wceq 1541  ifcif 4479  𝒫 cpw 4554  ∪ cuni 4863   class class class wbr 5098  ran crn 5625  ℩cio 6446   defAt wdfat 47362  ''''cafv2 47454

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-if 4480  df-afv2 47455

This theorem is referenced by:  dfatafv2ex  47459  funressndmafv2rn  47469  afv2eu  47484  afv2res  47485  tz6.12-afv2  47486  dfafv23  47499  rlimdmafv2  47504