Metamath Proof Explorer


Theorem isbasis3g

Description: Express the predicate "the set B is a basis for a topology". Definition of basis in Munkres p. 78. (Contributed by NM, 17-Jul-2006)

Ref Expression
Assertion isbasis3g ( 𝐵𝐶 → ( 𝐵 ∈ TopBases ↔ ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 isbasis2g ( 𝐵𝐶 → ( 𝐵 ∈ TopBases ↔ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) )
2 elssuni ( 𝑥𝐵𝑥 𝐵 )
3 2 rgen 𝑥𝐵 𝑥 𝐵
4 eluni2 ( 𝑥 𝐵 ↔ ∃ 𝑦𝐵 𝑥𝑦 )
5 4 biimpi ( 𝑥 𝐵 → ∃ 𝑦𝐵 𝑥𝑦 )
6 5 rgen 𝑥 𝐵𝑦𝐵 𝑥𝑦
7 3 6 pm3.2i ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 )
8 7 biantrur ( ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ↔ ( ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ) ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) )
9 df-3an ( ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) ↔ ( ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ) ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) )
10 8 9 bitr4i ( ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ↔ ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) )
11 1 10 bitrdi ( 𝐵𝐶 → ( 𝐵 ∈ TopBases ↔ ( ∀ 𝑥𝐵 𝑥 𝐵 ∧ ∀ 𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀ 𝑥𝐵𝑦𝐵𝑧 ∈ ( 𝑥𝑦 ) ∃ 𝑤𝐵 ( 𝑧𝑤𝑤 ⊆ ( 𝑥𝑦 ) ) ) ) )