Theorem pibp16 36882

Description: Property P000016 of pi-base. The class of compact topologies. A space 𝑋 is compact if every open cover of 𝑋 has a finite subcover. This theorem is just a relabeled copy of iscmp 23285. (Contributed by ML, 8-Dec-2020.)

Hypothesis

Ref	Expression
pibp16.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
pibp16	⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Distinct variable group:   𝑦,𝐽,𝑧

Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem pibp16

Step	Hyp	Ref	Expression
1		pibp16.x	. 2 ⊢ 𝑋 = ∪ 𝐽
2	1	iscmp 23285	1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ∩ cin 3943  𝒫 cpw 4598  ∪ cuni 4903  Fincfn 8957  Topctop 22788  Compccmp 23283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-in 3951  df-ss 3961  df-pw 4600  df-uni 4904  df-cmp 23284

This theorem is referenced by:  pibt1  36885