Theorem dfatafv2iota 44589

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 44588	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iftrue 4462	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
3	1, 2	syl5eq 2791	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))

Syntax hints:   → wi 4   = wceq 1539  ifcif 4456  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ran crn 5581  ℩cio 6374   defAt wdfat 44495  ''''cafv2 44587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457  df-afv2 44588

This theorem is referenced by:  dfatafv2ex  44592  funressndmafv2rn  44602  afv2eu  44617  afv2res  44618  tz6.12-afv2  44619  dfafv23  44632  rlimdmafv2  44637