Theorem pibp21 34699

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 4849	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		pibp21.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	syl6eqr 2874	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	pweqd 4558	. . . 4 ⊢ (𝑥 = 𝐽 → 𝒫 ∪ 𝑥 = 𝒫 𝑋)
5	4	difeq1d 4098	. . 3 ⊢ (𝑥 = 𝐽 → (𝒫 ∪ 𝑥 ∖ Fin) = (𝒫 𝑋 ∖ Fin))
6		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝐽 → (limPt‘𝑥) = (limPt‘𝐽))
7	6	fveq1d 6672	. . . . 5 ⊢ (𝑥 = 𝐽 → ((limPt‘𝑥)‘𝑦) = ((limPt‘𝐽)‘𝑦))
8	7	eleq2d 2898	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
9	3, 8	rexeqbidv 3402	. . 3 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦) ↔ ∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))
10	5, 9	raleqbidv 3401	. 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 3683	1 ⊢ (𝐽 ∈ 𝑊 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ (𝒫 𝑋 ∖ Fin)∃𝑧 ∈ 𝑋 𝑧 ∈ ((limPt‘𝐽)‘𝑦)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  {crab 3142   ∖ cdif 3933  𝒫 cpw 4539  ∪ cuni 4838  ‘cfv 6355  Fincfn 8509  Topctop 21501  limPtclp 21742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363

This theorem is referenced by:  pibt2  34701