Theorem fitop

Description: A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009)

	Ref	Expression
Assertion	fitop	$$⊢J\in \mathrm{Top}\to \mathrm{fi}\left(J\right)=J$$

Proof

Step	Hyp	Ref	Expression
1		inopn	$$⊢\left(J\in \mathrm{Top}\wedge x\in J\wedge y\in J\right)\to x\cap y\in J$$
2	1	3expib	$$⊢J\in \mathrm{Top}\to \left(\left(x\in J\wedge y\in J\right)\to x\cap y\in J\right)$$
3	2	ralrimivv	$$⊢J\in \mathrm{Top}\to \forall x\in J\forall y\in Jx\cap y\in J$$
4		inficl	$$⊢J\in \mathrm{Top}\to \left(\forall x\in J\forall y\in Jx\cap y\in J\leftrightarrow \mathrm{fi}\left(J\right)=J\right)$$
5	3 4	mpbid	$$⊢J\in \mathrm{Top}\to \mathrm{fi}\left(J\right)=J$$