Theorem pibp16 37355

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 23361. (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 23361	1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   ∩ cin 3932  𝒫 cpw 4582  ∪ cuni 4889  Fincfn 8968  Topctop 22866  Compccmp 23359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-ss 3950  df-pw 4584  df-uni 4890  df-cmp 23360

This theorem is referenced by:  pibt1  37358