Theorem pcmplfin 30525

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		simp2 1128	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ⊆ 𝐽)
2		ssexg 5041	. . . . . . 7 ⊢ ((𝑈 ⊆ 𝐽 ∧ 𝐽 ∈ Paracomp) → 𝑈 ∈ V)
3	2	ancoms 452	. . . . . 6 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽) → 𝑈 ∈ V)
4	3	3adant3 1123	. . . . 5 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ V)
5		elpwg 4387	. . . . 5 ⊢ (𝑈 ∈ V → (𝑈 ∈ 𝒫 𝐽 ↔ 𝑈 ⊆ 𝐽))
6	4, 5	syl 17	. . . 4 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → (𝑈 ∈ 𝒫 𝐽 ↔ 𝑈 ⊆ 𝐽))
7	1, 6	mpbird 249	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑈 ∈ 𝒫 𝐽)
8		ispcmp 30522	. . . . . 6 ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
9		pcmplfin.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
10	9	iscref 30509	. . . . . 6 ⊢ (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
11	8, 10	bitri 267	. . . . 5 ⊢ (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢)))
12	11	simprbi 492	. . . 4 ⊢ (𝐽 ∈ Paracomp → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
13	12	3ad2ant1 1124	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢))
14		simp3 1129	. . 3 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → 𝑋 = ∪ 𝑈)
15		unieq 4679	. . . . . 6 ⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪ 𝑈)
16	15	eqeq2d 2788	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑋 = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑈))
17		breq2 4890	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑣Ref𝑢 ↔ 𝑣Ref𝑈))
18	17	rexbidv 3237	. . . . 5 ⊢ (𝑢 = 𝑈 → (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢 ↔ ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈))
19	16, 18	imbi12d 336	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) ↔ (𝑋 = ∪ 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
20	19	rspcv 3507	. . 3 ⊢ (𝑈 ∈ 𝒫 𝐽 → (∀𝑢 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑢) → (𝑋 = ∪ 𝑈 → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)))
21	7, 13, 14, 20	syl3c 66	. 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈)
22		elin 4019	. . . . 5 ⊢ (𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ 𝑣 ∈ (LocFin‘𝐽)))
23	22	anbi1i 617	. . . 4 ⊢ ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ ((𝑣 ∈ 𝒫 𝐽 ∧ 𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈))
24		anass 462	. . . 4 ⊢ (((𝑣 ∈ 𝒫 𝐽 ∧ 𝑣 ∈ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
25	23, 24	bitri 267	. . 3 ⊢ ((𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑣Ref𝑈) ↔ (𝑣 ∈ 𝒫 𝐽 ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)))
26	25	rexbii2 3222	. 2 ⊢ (∃𝑣 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑣Ref𝑈 ↔ ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
27	21, 26	sylib 210	1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  𝒫 cpw 4379  ∪ cuni 4671   class class class wbr 4886  ‘cfv 6135  Topctop 21105  Refcref 21714  LocFinclocfin 21716  CovHasRefccref 30507  Paracompcpcmp 30520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-cref 30508  df-pcmp 30521

This theorem is referenced by:  pcmplfinf  30526