Metamath Proof Explorer


Theorem basis1

Description: Property of a basis. (Contributed by NM, 16-Jul-2006)

Ref Expression
Assertion basis1 ( ( 𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵 ) → ( 𝐶𝐷 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 isbasisg ( 𝐵 ∈ TopBases → ( 𝐵 ∈ TopBases ↔ ∀ 𝑥𝐵𝑦𝐵 ( 𝑥𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) ) )
2 1 ibi ( 𝐵 ∈ TopBases → ∀ 𝑥𝐵𝑦𝐵 ( 𝑥𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) )
3 ineq1 ( 𝑥 = 𝐶 → ( 𝑥𝑦 ) = ( 𝐶𝑦 ) )
4 3 pweqd ( 𝑥 = 𝐶 → 𝒫 ( 𝑥𝑦 ) = 𝒫 ( 𝐶𝑦 ) )
5 4 ineq2d ( 𝑥 = 𝐶 → ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) )
6 5 unieqd ( 𝑥 = 𝐶 ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) )
7 3 6 sseq12d ( 𝑥 = 𝐶 → ( ( 𝑥𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) ↔ ( 𝐶𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) ) )
8 ineq2 ( 𝑦 = 𝐷 → ( 𝐶𝑦 ) = ( 𝐶𝐷 ) )
9 8 pweqd ( 𝑦 = 𝐷 → 𝒫 ( 𝐶𝑦 ) = 𝒫 ( 𝐶𝐷 ) )
10 9 ineq2d ( 𝑦 = 𝐷 → ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) )
11 10 unieqd ( 𝑦 = 𝐷 ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) = ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) )
12 8 11 sseq12d ( 𝑦 = 𝐷 → ( ( 𝐶𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝑦 ) ) ↔ ( 𝐶𝐷 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) ) )
13 7 12 rspc2v ( ( 𝐶𝐵𝐷𝐵 ) → ( ∀ 𝑥𝐵𝑦𝐵 ( 𝑥𝑦 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝑥𝑦 ) ) → ( 𝐶𝐷 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) ) )
14 2 13 syl5com ( 𝐵 ∈ TopBases → ( ( 𝐶𝐵𝐷𝐵 ) → ( 𝐶𝐷 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) ) )
15 14 3impib ( ( 𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵 ) → ( 𝐶𝐷 ) ⊆ ( 𝐵 ∩ 𝒫 ( 𝐶𝐷 ) ) )