Theorem kgenuni

Description: The base set of the compact generator is the same as the original topology. (Contributed by Mario Carneiro, 20-Mar-2015)

	Ref	Expression
Hypothesis	kgenuni.1	⊢ 𝑋 = ∪ 𝐽
Assertion	kgenuni	⊢ ( 𝐽 ∈ Top → 𝑋 = ∪ ( 𝑘Gen ‘ 𝐽 ) )

Proof

Step	Hyp	Ref	Expression
1		kgenuni.1	⊢ 𝑋 = ∪ 𝐽
2	1	toptopon	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		kgentopon	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ 𝑋 ) )
4	2 3	sylbi	⊢ ( 𝐽 ∈ Top → ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ 𝑋 ) )
5		toponuni	⊢ ( ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ ( 𝑘Gen ‘ 𝐽 ) )
6	4 5	syl	⊢ ( 𝐽 ∈ Top → 𝑋 = ∪ ( 𝑘Gen ‘ 𝐽 ) )