Theorem stoig

Description: The topological space built with a subspace topology. (Contributed by FL, 5-Jan-2009) (Proof shortened by Mario Carneiro, 1-May-2015)

	Ref	Expression
Hypothesis	restuni.1	$$⊢X=\bigcup J$$
Assertion	stoig	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J{\upharpoonright }_{𝑡}A⟩\right\}\in \mathrm{TopSp}$$

Proof

Step	Hyp	Ref	Expression
1		restuni.1	$$⊢X=\bigcup J$$
2	1	toptopon	$$⊢J\in \mathrm{Top}\leftrightarrow J\in \mathrm{TopOn}\left(X\right)$$
3		resttopon	$$⊢\left(J\in \mathrm{TopOn}\left(X\right)\wedge A\subseteq X\right)\to J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)$$
4	2 3	sylanb	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)$$
5		eqid	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J{\upharpoonright }_{𝑡}A⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J{\upharpoonright }_{𝑡}A⟩\right\}$$
6	5	eltpsg	$$⊢J{\upharpoonright }_{𝑡}A\in \mathrm{TopOn}\left(A\right)\to \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J{\upharpoonright }_{𝑡}A⟩\right\}\in \mathrm{TopSp}$$
7	4 6	syl	$$⊢\left(J\in \mathrm{Top}\wedge A\subseteq X\right)\to \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},A⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J{\upharpoonright }_{𝑡}A⟩\right\}\in \mathrm{TopSp}$$