Theorem pibp16 38168

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 23614. (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 23614	1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898  𝒫 cpw 4557  ∪ cuni 4867  Fincfn 8953  Topctop 23119  Compccmp 23612

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3916  df-pw 4559  df-uni 4868  df-cmp 23613

This theorem is used by:  pibt1  38171