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Theorem pibp19 34826
Description: Property P000019 of pi-base. The class of countably compact topologies. A space 𝑋 is countably compact if every countable open cover of 𝑋 has a finite subcover. (Contributed by ML, 8-Dec-2020.)
Hypotheses
Ref Expression
pibp19.x 𝑋 = 𝐽
pibp19.19 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
Assertion
Ref Expression
pibp19 (𝐽𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Distinct variable groups:   𝑥,𝐽,𝑦   𝑧,𝐽,𝑥   𝑥,𝑋
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)   𝑋(𝑦,𝑧)

Proof of Theorem pibp19
StepHypRef Expression
1 pweq 4516 . . 3 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2 unieq 4814 . . . . . . 7 (𝑥 = 𝐽 𝑥 = 𝐽)
3 pibp19.x . . . . . . 7 𝑋 = 𝐽
42, 3eqtr4di 2854 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝑋)
54eqeq1d 2803 . . . . 5 (𝑥 = 𝐽 → ( 𝑥 = 𝑦𝑋 = 𝑦))
65anbi1d 632 . . . 4 (𝑥 = 𝐽 → (( 𝑥 = 𝑦𝑦 ≼ ω) ↔ (𝑋 = 𝑦𝑦 ≼ ω)))
74eqeq1d 2803 . . . . 5 (𝑥 = 𝐽 → ( 𝑥 = 𝑧𝑋 = 𝑧))
87rexbidv 3259 . . . 4 (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
96, 8imbi12d 348 . . 3 (𝑥 = 𝐽 → ((( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ ((𝑋 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
101, 9raleqbidv 3357 . 2 (𝑥 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑥(( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
11 pibp19.19 . 2 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(( 𝑥 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
1210, 11elrab2 3634 1 (𝐽𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = 𝑦𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  cin 3883  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  ωcom 7564  cdom 8494  Fincfn 8496  Topctop 21501
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804
This theorem is referenced by:  pibt1  34828  pibt2  34829
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