Theorem pibp16 38087

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 23554. (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 23554	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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3904  𝒫 cpw 4562  ∪ cuni 4872  Fincfn 8939  Topctop 23059  Compccmp 23552

This proof depends on 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 proof 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  df-cmp 23553

This theorem is used by:  pibt1  38090