Theorem tg1

Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006)

	Ref	Expression
Assertion	tg1	⊢ ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐴 ⊆ ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elfvdm	⊢ ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐵 ∈ dom topGen )
2		eltg2	⊢ ( 𝐵 ∈ dom topGen → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
3	2	simprbda	⊢ ( ( 𝐵 ∈ dom topGen ∧ 𝐴 ∈ ( topGen ‘ 𝐵 ) ) → 𝐴 ⊆ ∪ 𝐵 )
4	1 3	mpancom	⊢ ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐴 ⊆ ∪ 𝐵 )