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Theorem pibt1 34778
Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 34775 and pibp19 34776 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 pm3.41 496 . . . 4 (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) → (( 𝐽 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
21ralimi 3155 . . 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 619 . 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 2824 . . 3 𝐽 = 𝐽
54pibp16 34775 . 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 34776 . 2 (𝐽𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(( 𝐽 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
83, 5, 73imtr4i 295 1 (𝐽 ∈ Comp → 𝐽𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wrex 3134  {crab 3137  cin 3918  𝒫 cpw 4522   cuni 4824   class class class wbr 5052  ωcom 7574  cdom 8503  Fincfn 8505  Topctop 21501  Compccmp 21994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-uni 4825  df-cmp 21995
This theorem is referenced by: (None)
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