Theorem tpsuni

Description: The base set of a topological space. (Contributed by FL, 27-Jun-2014)

	Ref	Expression
Hypotheses	istps.a	⊢ 𝐴 = ( Base ‘ 𝐾 )
	istps.j	⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
Assertion	tpsuni	⊢ ( 𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽 )

Proof

Step	Hyp	Ref	Expression
1		istps.a	⊢ 𝐴 = ( Base ‘ 𝐾 )
2		istps.j	⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
3	1 2	istps2	⊢ ( 𝐾 ∈ TopSp ↔ ( 𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽 ) )
4	3	simprbi	⊢ ( 𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽 )