Theorem topgrpbas

Description: The base set 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	topgrpbas	$$⊢B\in X\to B={\mathrm{Base}}_W$$

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		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		snsstp1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq W$$
6	2 3 5	strfv	$$⊢B\in X\to B={\mathrm{Base}}_W$$