Theorem fabex

Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007)

	Ref	Expression
Hypotheses	fabex.1	$$⊢A\in \mathrm{V}$$
	fabex.2	$$⊢B\in \mathrm{V}$$
	fabex.3	$$⊢F=\left\{x|\left(x:A⟶B\wedge \phi \right)\right\}$$
Assertion	fabex	$$⊢F\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		fabex.1	$$⊢A\in \mathrm{V}$$
2		fabex.2	$$⊢B\in \mathrm{V}$$
3		fabex.3	$$⊢F=\left\{x|\left(x:A⟶B\wedge \phi \right)\right\}$$
4	3	fabexg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to F\in \mathrm{V}$$
5	1 2 4	mp2an	$$⊢F\in \mathrm{V}$$