Theorem pibp21 37393

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	⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 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

Step	Hyp	Ref	Expression
1		unieq 4869	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		pibp21.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	pweqd 4568	. . . 4 ⊢ (𝑥 = 𝐽 → 𝒫 ∪ 𝑥 = 𝒫 𝑋)
5	4	difeq1d 4076	. . 3 ⊢ (𝑥 = 𝐽 → (𝒫 ∪ 𝑥 ∖ Fin) = (𝒫 𝑋 ∖ Fin))
6		fveq2 6822	. . . . . 6 ⊢ (𝑥 = 𝐽 → (limPt‘𝑥) = (limPt‘𝐽))
7	6	fveq1d 6824	. . . . 5 ⊢ (𝑥 = 𝐽 → ((limPt‘𝑥)‘𝑦) = ((limPt‘𝐽)‘𝑦))
8	7	eleq2d 2814	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
9	3, 8	rexeqbidv 3310	. . 3 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
10	5, 9	raleqbidv 3309	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
11		pibp21.21	. 2 ⊢ 𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}
12	10, 11	elrab2 3651	1 ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394   ∖ cdif 3900  𝒫 cpw 4551  ∪ cuni 4858  ‘cfv 6482  Fincfn 8872  Topctop 22778  limPtclp 23019

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490

This theorem is referenced by:  pibt2  37395