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Metamath Proof Explorer

Theorem ist0-2

Description: The predicate "is a T_0 space". (Contributed by Mario Carneiro, 24-Aug-2015)

Ref Expression
Assertion ist0-2 ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 topontop ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top )
2 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
3 2 ist0 ⊢ ( 𝐽 ∈ Kol2 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
4 3 baib ⊢ ( 𝐽 ∈ Top → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
5 1 4 syl ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
6 toponuni ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 )
7 6 raleqdv ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
8 6 7 raleqbidv ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∈ ∪ 𝐽 ∀ 𝑦 ∈ ∪ 𝐽 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )
9 5 8 bitr4d ⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐽 ∈ Kol2 ↔ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ∀ 𝑜 ∈ 𝐽 ( 𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜 ) → 𝑥 = 𝑦 ) ) )