Metamath Proof Explorer


Theorem pibt1

Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 and pibp19 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020)

Ref Expression
Hypothesis pibt1.19 𝐶 = { 𝑥 ∈ Top ∣ ∀ 𝑦 ∈ 𝒫 𝑥 ( ( 𝑥 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝑥 = 𝑧 ) }
Assertion pibt1 ( 𝐽 ∈ Comp → 𝐽𝐶 )

Proof

Step Hyp Ref Expression
1 pibt1.19 𝐶 = { 𝑥 ∈ Top ∣ ∀ 𝑦 ∈ 𝒫 𝑥 ( ( 𝑥 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝑥 = 𝑧 ) }
2 pm3.41 ( ( 𝐽 = 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) → ( ( 𝐽 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) )
3 2 ralimi ( ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝐽 = 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) → ∀ 𝑦 ∈ 𝒫 𝐽 ( ( 𝐽 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) )
4 3 anim2i ( ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝐽 = 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) ) → ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ( 𝐽 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) ) )
5 eqid 𝐽 = 𝐽
6 5 pibp16 ( 𝐽 ∈ Comp ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( 𝐽 = 𝑦 → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) ) )
7 5 1 pibp19 ( 𝐽𝐶 ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝐽 ( ( 𝐽 = 𝑦𝑦 ≼ ω ) → ∃ 𝑧 ∈ ( 𝒫 𝑦 ∩ Fin ) 𝐽 = 𝑧 ) ) )
8 4 6 7 3imtr4i ( 𝐽 ∈ Comp → 𝐽𝐶 )