Metamath Proof Explorer


Theorem topgrptset

Description: The topology 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 topgrptset ( 𝐽𝑋𝐽 = ( TopSet ‘ 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 tsetid TopSet = Slot ( TopSet ‘ ndx )
4 snsstp3 { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ }
5 4 1 sseqtrri { ⟨ ( TopSet ‘ ndx ) , 𝐽 ⟩ } ⊆ 𝑊
6 2 3 5 strfv ( 𝐽𝑋𝐽 = ( TopSet ‘ 𝑊 ) )