Theorem eltg2b

Description: Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014) (Revised by Mario Carneiro, 10-Jan-2015)

	Ref	Expression
Assertion	eltg2b	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eltg2	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) ) )
2		simpl	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) → 𝑥 ∈ 𝑦 )
3	2	reximi	⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) → ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )
4		eluni2	⊢ ( 𝑥 ∈ ∪ 𝐵 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 )
5	3 4	sylibr	⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) → 𝑥 ∈ ∪ 𝐵 )
6	5	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 )
7		dfss3	⊢ ( 𝐴 ⊆ ∪ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝐵 )
8	6 7	sylibr	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) → 𝐴 ⊆ ∪ 𝐵 )
9	8	pm4.71ri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )
10	1 9	bitr4di	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴 ) ) )