Theorem pcmplfin 34540

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 5280	. . . . . 6 ⊢ ((𝑈 ⊆ 𝐽 ∧ 𝐽 ∈ Paracomp) → 𝑈 ∈ V)
2	1	ancoms 464	. . . . 5 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽) → 𝑈 ∈ V)
3	2	3adant3 1150	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ V)
4		simp2 1155	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ⊆ 𝐽)
5	3, 4	elpwd 4562	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ 𝒫 𝐽)
6		ispcmp 34537	. . . . . 6 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
7		pcmplfin.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
8	7	iscref 34524	. . . . . 6 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
9	6, 8	bitri 278	. . . . 5 ⊢ (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
10	9	simprbi 503	. . . 4 ⊢ (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
11	10	3ad2ant1 1151	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
12		simp3 1156	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑋 = ∪ 𝑈)
13		unieq 4877	. . . . . 6 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
14	13	eqeq2d 2771	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑋 = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑈))
15		breq2 5106	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑣Ref𝑢 ↔ 𝑣Ref𝑈))
16	15	rexbidv 3186	. . . . 5 ⊢ (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
17	14, 16	imbi12d 347	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = ∪ 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
18	17	rspcv 3572	. . 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 67	. 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)
20		rexin 4146	. 2 ⊢ (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
21	19, 20	sylib 221	1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  ‘cfv 6541  Topctop 23218  Refcref 23828  LocFinclocfin 23830  CovHasRefccref 34522  Paracompcpcmp 34535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-cref 34523  df-pcmp 34536

This theorem is used by:  pcmplfinf  34541