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Theorem pibp21 34720
Description: Property P000021 of pi-base. The class of weakly countably compact topologies, or limit point compact topologies. A space 𝑋 is weakly countably compact if every infinite subset of 𝑋 has a limit point. (Contributed by ML, 9-Dec-2020.)
Hypotheses
Ref Expression
pibp21.x 𝑋 = 𝐽
pibp21.21 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 𝑥 ∖ Fin)∃𝑧 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}
Assertion
Ref Expression
pibp21 (𝐽𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
Distinct variable groups:   𝑥,𝐽,𝑦   𝑧,𝐽,𝑥   𝑥,𝑋,𝑦   𝑧,𝑋
Allowed substitution hints:   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem pibp21
StepHypRef Expression
1 unieq 4822 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
2 pibp21.x . . . . . 6 𝑋 = 𝐽
31, 2syl6eqr 2874 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
43pweqd 4531 . . . 4 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝑋)
54difeq1d 4074 . . 3 (𝑥 = 𝐽 → (𝒫 𝑥 ∖ Fin) = (𝒫 𝑋 ∖ Fin))
6 fveq2 6643 . . . . . 6 (𝑥 = 𝐽 → (limPt‘𝑥) = (limPt‘𝐽))
76fveq1d 6645 . . . . 5 (𝑥 = 𝐽 → ((limPt‘𝑥)‘𝑦) = ((limPt‘𝐽)‘𝑦))
87eleq2d 2897 . . . 4 (𝑥 = 𝐽 → (𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
93, 8rexeqbidv 3387 . . 3 (𝑥 = 𝐽 → (∃𝑧 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∃𝑧𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
105, 9raleqbidv 3386 . 2 (𝑥 = 𝐽 → (∀𝑦 ∈ (𝒫 𝑥 ∖ Fin)∃𝑧 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
11 pibp21.21 . 2 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 𝑥 ∖ Fin)∃𝑧 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}
1210, 11elrab2 3660 1 (𝐽𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wrex 3127  {crab 3130  cdif 3907  𝒫 cpw 4512   cuni 4811  cfv 6328  Fincfn 8484  Topctop 21477  limPtclp 21718
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 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336
This theorem is referenced by:  pibt2  34722
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