Theorem toptopon

Description: Alternative definition of Top in terms of TopOn . (Contributed by Mario Carneiro, 13-Aug-2015)

	Ref	Expression
Hypothesis	toptopon.1	⊢ 𝑋 = ∪ 𝐽
Assertion	toptopon	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		toptopon.1	⊢ 𝑋 = ∪ 𝐽
2		istopon	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ ( 𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽 ) )
3	1 2	mpbiran2	⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ↔ 𝐽 ∈ Top )
4	3	bicomi	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )