Theorem pibp19 38080

Description: Property P000019 of pi-base. The class of countably compact topologies. A space 𝑋 is countably compact if every countable open cover of 𝑋 has a finite subcover. (Contributed by ML, 8-Dec-2020.)

Hypotheses

Ref	Expression
pibp19.x	⊢ 𝑋 = ∪ 𝐽
pibp19.19	⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}

Assertion

Ref	Expression
pibp19	⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Distinct variable groups:   𝑥,𝐽,𝑦   𝑧,𝐽,𝑥   𝑥,𝑋

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)   𝑋(𝑦,𝑧)

Proof of Theorem pibp19

Step	Hyp	Ref	Expression
1		pweq 4576	. . 3 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2		unieq 4883	. . . . . . 7 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
3		pibp19.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	eqtr4di 2816	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
5	4	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	5	anbi1d 642	. . . 4 ⊢ (𝑥 = 𝐽 → ((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) ↔ (𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω)))
7	4	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝐽 → (∪ 𝑥 = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
8	7	rexbidv 3189	. . . 4 ⊢ (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
9	6, 8	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐽 → (((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ ((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
10	1, 9	raleqbidv 3338	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
11		pibp19.19	. 2 ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
12	10, 11	elrab2 3654	1 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((𝑋 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416   ∩ cin 3904  𝒫 cpw 4562  ∪ cuni 4872   class class class wbr 5109  ωcom 7858   ≼ cdom 8937  Fincfn 8939  Topctop 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-ss 3922  df-pw 4564  df-uni 4873

This theorem is referenced by:  pibt1  38082  pibt2  38083