Theorem dfatafv2iota

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)

	Ref	Expression
Assertion	dfatafv2iota	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\left(\iota x|AFx\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-afv2	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=if\left(F\mathrm{defAt}A,\left(\iota x|AFx\right),𝒫\bigcup \mathrm{ran}F\right)$$
2		iftrue	$$⊢F\mathrm{defAt}A\to if\left(F\mathrm{defAt}A,\left(\iota x|AFx\right),𝒫\bigcup \mathrm{ran}F\right)=\left(\iota x|AFx\right)$$
3	1 2	syl5eq	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\left(\iota x|AFx\right)$$