Theorem eufsn2

Description: There is exactly one function into a singleton, assuming ax-pow and ax-un . Variant of eufsn . If existence is not needed, use mofsn or mofsn2 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024)

	Ref	Expression
Hypotheses	eufsn.1	$$⊢\phi \to B\in W$$
	eufsn.2	$$⊢\phi \to A\in V$$
Assertion	eufsn2	$$⊢\phi \to \exists !ff:A⟶\left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		eufsn.1	$$⊢\phi \to B\in W$$
2		eufsn.2	$$⊢\phi \to A\in V$$
3		snex	$$⊢\left\{B\right\}\in \mathrm{V}$$
4		xpexg	$$⊢\left(A\in V\wedge \left\{B\right\}\in \mathrm{V}\right)\to A\times \left\{B\right\}\in \mathrm{V}$$
5	2 3 4	sylancl	$$⊢\phi \to A\times \left\{B\right\}\in \mathrm{V}$$
6	1 5	eufsnlem	$$⊢\phi \to \exists !ff:A⟶\left\{B\right\}$$