Theorem pibp21 37857

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 4870	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		pibp21.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2809	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	pweqd 4566	. . . 4 ⊢ (𝑥 = 𝐽 → 𝒫 ∪ 𝑥 = 𝒫 𝑋)
5	4	difeq1d 4074	. . 3 ⊢ (𝑥 = 𝐽 → (𝒫 ∪ 𝑥 ∖ Fin) = (𝒫 𝑋 ∖ Fin))
6		fveq2 6856	. . . . . 6 ⊢ (𝑥 = 𝐽 → (limPt‘𝑥) = (limPt‘𝐽))
7	6	fveq1d 6858	. . . . 5 ⊢ (𝑥 = 𝐽 → ((limPt‘𝑥)‘𝑦) = ((limPt‘𝐽)‘𝑦))
8	7	eleq2d 2842	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
9	3, 8	rexeqbidv 3331	. . 3 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
10	5, 9	raleqbidv 3330	. 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 3648	1 ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∀wral 3070  ∃wrex 3080  {crab 3408   ∖ cdif 3896  𝒫 cpw 4549  ∪ cuni 4859  ‘cfv 6510  Fincfn 8916  Topctop 22926  limPtclp 23167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-iota 6466  df-fv 6518

This theorem is referenced by:  pibt2  37859