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Theorem pibt1 36602
Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 36599 and pibp19 36600 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.)
Hypothesis
Ref Expression
pibt1.19 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
Assertion
Ref Expression
pibt1 (𝐽 ∈ Comp → 𝐽𝐶)
Distinct variable group:   𝑥,𝐽,𝑦,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem pibt1
StepHypRef Expression
1 pm3.41 491 . . . 4 (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) → (( 𝐽 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
21ralimi 3081 . . 3 (∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) → ∀𝑦 ∈ 𝒫 𝐽(( 𝐽 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
32anim2i 615 . 2 ((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)))
4 eqid 2730 . . 3 𝐽 = 𝐽
54pibp16 36599 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
6 pibt1.19 . . 3 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
74, 6pibp19 36600 . 2 (𝐽𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(( 𝐽 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
83, 5, 73imtr4i 291 1 (𝐽 ∈ Comp → 𝐽𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  wrex 3068  {crab 3430  cin 3948  𝒫 cpw 4603   cuni 4909   class class class wbr 5149  ωcom 7859  cdom 8941  Fincfn 8943  Topctop 22617  Compccmp 23112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-cmp 23113
This theorem is referenced by: (None)
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