Theorem pibp21 37909

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 4876	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		pibp21.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2815	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	pweqd 4572	. . . 4 ⊢ (𝑥 = 𝐽 → 𝒫 ∪ 𝑥 = 𝒫 𝑋)
5	4	difeq1d 4079	. . 3 ⊢ (𝑥 = 𝐽 → (𝒫 ∪ 𝑥 ∖ Fin) = (𝒫 𝑋 ∖ Fin))
6		fveq2 6867	. . . . . 6 ⊢ (𝑥 = 𝐽 → (limPt‘𝑥) = (limPt‘𝐽))
7	6	fveq1d 6869	. . . . 5 ⊢ (𝑥 = 𝐽 → ((limPt‘𝑥)‘𝑦) = ((limPt‘𝐽)‘𝑦))
8	7	eleq2d 2848	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
9	3, 8	rexeqbidv 3337	. . 3 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
10	5, 9	raleqbidv 3336	. 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 3654	1 ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414   ∖ cdif 3901  𝒫 cpw 4555  ∪ cuni 4865  ‘cfv 6521  Fincfn 8927  Topctop 22953  limPtclp 23194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529

This theorem is referenced by:  pibt2  37911