Theorem restfn

Description: The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015)

	Ref	Expression
Assertion	restfn	$$⊢{\upharpoonright }_{𝑡}Fn\left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-rest	$$⊢{\upharpoonright }_{𝑡}=\left(j\in \mathrm{V},x\in \mathrm{V}⟼\mathrm{ran}\left(y\in j⟼y\cap x\right)\right)$$
2		vex	$$⊢j\in \mathrm{V}$$
3	2	mptex	$$⊢\left(y\in j⟼y\cap x\right)\in \mathrm{V}$$
4	3	rnex	$$⊢\mathrm{ran}\left(y\in j⟼y\cap x\right)\in \mathrm{V}$$
5	1 4	fnmpoi	$$⊢{\upharpoonright }_{𝑡}Fn\left(\mathrm{V}\times \mathrm{V}\right)$$