Theorem pibt1 38082

Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 38079 and pibp19 38080 for the definitions of the relevant properties. (Contributed by ML, 8-Dec-2020.)

Hypothesis

Ref	Expression
pibt1.19	⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}

Assertion

Ref	Expression
pibt1	⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

Distinct variable group:   𝑥,𝐽,𝑦,𝑧

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

Proof of Theorem pibt1

Step	Hyp	Ref	Expression
1		pm3.41 497	. . . 4 ⊢ ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) → ((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
2	1	ralimi 3102	. . 3 ⊢ (∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) → ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
3	2	anim2i 628	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)) → (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
4		eqid 2763	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	pibp16 38079	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
6		pibt1.19	. . 3 ⊢ 𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}
7	4, 6	pibp19 38080	. 2 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
8	3, 5, 7	3imtr4i 295	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ 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  Compccmp 23543

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  df-cmp 23544

This theorem is referenced by: (None)