Theorem elpclN

Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclfval.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	pclfval.s	$$⊢S=\mathrm{PSubSp}\left(K\right)$$
	pclfval.c	$$⊢U=\mathrm{PCl}\left(K\right)$$
	elpcl.q	$$⊢Q\in \mathrm{V}$$
Assertion	elpclN	$$⊢\left(K\in V\wedge X\subseteq A\right)\to \left(Q\in U\left(X\right)\leftrightarrow \forall y\in S\left(X\subseteq y\to Q\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pclfval.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		pclfval.s	$$⊢S=\mathrm{PSubSp}\left(K\right)$$
3		pclfval.c	$$⊢U=\mathrm{PCl}\left(K\right)$$
4		elpcl.q	$$⊢Q\in \mathrm{V}$$
5	1 2 3	pclvalN	$$⊢\left(K\in V\wedge X\subseteq A\right)\to U\left(X\right)=\bigcap \left\{y\in S|X\subseteq y\right\}$$
6	5	eleq2d	$$⊢\left(K\in V\wedge X\subseteq A\right)\to \left(Q\in U\left(X\right)\leftrightarrow Q\in \bigcap \left\{y\in S|X\subseteq y\right\}\right)$$
7	4	elintrab	$$⊢Q\in \bigcap \left\{y\in S|X\subseteq y\right\}\leftrightarrow \forall y\in S\left(X\subseteq y\to Q\in y\right)$$
8	6 7	bitrdi	$$⊢\left(K\in V\wedge X\subseteq A\right)\to \left(Q\in U\left(X\right)\leftrightarrow \forall y\in S\left(X\subseteq y\to Q\in y\right)\right)$$