Theorem dfatafv2iota 48007

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 48006	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
2		iftrue 4495	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = (℩𝑥𝐴𝐹𝑥))
3	1, 2	eqtrid 2812	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ifcif 4489  𝒫 cpw 4564  ∪ cuni 4874   class class class wbr 5111  ran crn 5664  ℩cio 6494   defAt wdfat 47913  ''''cafv2 48005

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-if 4490  df-afv2 48006

This theorem is used by:  dfatafv2ex  48010  funressndmafv2rn  48020  afv2eu  48035  afv2res  48036  tz6.12-afv2  48037  dfafv23  48050  rlimdmafv2  48055