Theorem topgrptset

Description: The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	topgrpfn.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
Assertion	topgrptset	$$⊢J\in X\to J=\mathrm{TopSet}\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		topgrpfn.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
2	1	topgrpstr	$$⊢W\mathrm{Struct}⟨1,9⟩$$
3		tsetid	$$⊢\mathrm{TopSet}=\mathrm{Slot}\mathrm{TopSet}\left(\mathrm{ndx}\right)$$
4		snsstp3	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}\subseteq W$$
6	2 3 5	strfv	$$⊢J\in X\to J=\mathrm{TopSet}\left(W\right)$$