Theorem pibt1 36297

Description: Theorem T000001 of pi-base. A compact topology is also countably compact. See pibp16 36294 and pibp19 36295 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		pm3.41 494	. . . 4 ⊢ ((∪ 𝐽 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧) → ((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧))
2	1	ralimi 3084	. . 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 618	. 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 2733	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
5	4	pibp16 36294	. 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 36295	. 2 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽((∪ 𝐽 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝐽 = ∪ 𝑧)))
8	3, 5, 7	3imtr4i 292	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433   ∩ cin 3948  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  ωcom 7855   ≼ cdom 8937  Fincfn 8939  Topctop 22395  Compccmp 22890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-cmp 22891

This theorem is referenced by: (None)