Theorem tg2

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

	Ref	Expression
Assertion	tg2	⊢ ( ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ∧ 𝐶 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elfvdm	⊢ ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → 𝐵 ∈ dom topGen )
2		eltg2b	⊢ ( 𝐵 ∈ dom topGen → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ↔ ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ 𝐵 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
3		eleq1	⊢ ( 𝑦 = 𝐶 → ( 𝑦 ∈ 𝑥 ↔ 𝐶 ∈ 𝑥 ) )
4	3	anbi1d	⊢ ( 𝑦 = 𝐶 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
5	4	rexbidv	⊢ ( 𝑦 = 𝐶 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
6	5	rspccv	⊢ ( ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ 𝐵 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) → ( 𝐶 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
7	2 6	syl6bi	⊢ ( 𝐵 ∈ dom topGen → ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → ( 𝐶 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) ) )
8	1 7	mpcom	⊢ ( 𝐴 ∈ ( topGen ‘ 𝐵 ) → ( 𝐶 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) ) )
9	8	imp	⊢ ( ( 𝐴 ∈ ( topGen ‘ 𝐵 ) ∧ 𝐶 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐵 ( 𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴 ) )