Theorem pcmplfin 31712

Description: Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.)

Hypothesis

Ref	Expression
pcmplfin.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
pcmplfin	⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))

Distinct variable groups:   𝑣,𝐽   𝑣,𝑈

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem pcmplfin

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssexg 5242	. . . . . 6 ⊢ ((𝑈 ⊆ 𝐽 ∧ 𝐽 ∈ Paracomp) → 𝑈 ∈ V)
2	1	ancoms 458	. . . . 5 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽) → 𝑈 ∈ V)
3	2	3adant3 1130	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ V)
4		simp2 1135	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ⊆ 𝐽)
5	3, 4	elpwd 4538	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ 𝒫 𝐽)
6		ispcmp 31709	. . . . . 6 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
7		pcmplfin.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
8	7	iscref 31696	. . . . . 6 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
9	6, 8	bitri 274	. . . . 5 ⊢ (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
10	9	simprbi 496	. . . 4 ⊢ (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
11	10	3ad2ant1 1131	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
12		simp3 1136	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑋 = ∪ 𝑈)
13		unieq 4847	. . . . . 6 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
14	13	eqeq2d 2749	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑋 = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑈))
15		breq2 5074	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑣Ref𝑢 ↔ 𝑣Ref𝑈))
16	15	rexbidv 3225	. . . . 5 ⊢ (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
17	14, 16	imbi12d 344	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = ∪ 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
18	17	rspcv 3547	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐽 → (∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) → (𝑋 = ∪ 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
19	5, 11, 12, 18	syl3c 66	. 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)
20		rexin 4170	. 2 ⊢ (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
21	19, 20	sylib 217	1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418  Topctop 21950  Refcref 22561  LocFinclocfin 22563  CovHasRefccref 31694  Paracompcpcmp 31707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-cref 31695  df-pcmp 31708

This theorem is referenced by:  pcmplfinf  31713