Theorem topgrpbas

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

	Ref	Expression
Hypothesis	topgrpfn.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ }
Assertion	topgrpbas	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		topgrpfn.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ }
2	1	topgrpstr	⊢ 𝑊 Struct ⟨ 1 , 9 ⟩
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ }
5	4 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝑊
6	2 3 5	strfv	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝑊 ) )