Theorem pcmplfin 29927

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 = U. J

Assertion

Ref	Expression
pcmplfin	|- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> E. v e. ~P J ( v e. ( LocFin ` J ) /\ v Ref U ) )

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 1062	. . . 4 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> U C_ J )
2		ssexg 4804	. . . . . . 7 |- ( ( U C_ J /\ J e. Paracomp ) -> U e. _V )
3	2	ancoms 469	. . . . . 6 |- ( ( J e. Paracomp /\ U C_ J ) -> U e. _V )
4	3	3adant3 1081	. . . . 5 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> U e. _V )
5		elpwg 4166	. . . . 5 |- ( U e. _V -> ( U e. ~P J <-> U C_ J ) )
6	4, 5	syl 17	. . . 4 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> ( U e. ~P J <-> U C_ J ) )
7	1, 6	mpbird 247	. . 3 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> U e. ~P J )
8		ispcmp 29924	. . . . . 6 |- ( J e. Paracomp <-> J e. CovHasRef ( LocFin ` J ) )
9		pcmplfin.x	. . . . . . 7 |- X = U. J
10	9	iscref 29911	. . . . . 6 |- ( J e. CovHasRef ( LocFin ` J ) <-> ( J e. Top /\ A. u e. ~P J ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) ) )
11	8, 10	bitri 264	. . . . 5 |- ( J e. Paracomp <-> ( J e. Top /\ A. u e. ~P J ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) ) )
12	11	simprbi 480	. . . 4 |- ( J e. Paracomp -> A. u e. ~P J ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) )
13	12	3ad2ant1 1082	. . 3 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> A. u e. ~P J ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) )
14		simp3 1063	. . 3 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> X = U. U )
15		unieq 4444	. . . . . 6 |- ( u = U -> U. u = U. U )
16	15	eqeq2d 2632	. . . . 5 |- ( u = U -> ( X = U. u <-> X = U. U ) )
17		breq2 4657	. . . . . 6 |- ( u = U -> ( v Ref u <-> v Ref U ) )
18	17	rexbidv 3052	. . . . 5 |- ( u = U -> ( E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u <-> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref U ) )
19	16, 18	imbi12d 334	. . . 4 |- ( u = U -> ( ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) <-> ( X = U. U -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref U ) ) )
20	19	rspcv 3305	. . 3 |- ( U e. ~P J -> ( A. u e. ~P J ( X = U. u -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref u ) -> ( X = U. U -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref U ) ) )
21	7, 13, 14, 20	syl3c 66	. 2 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref U )
22		elin 3796	. . . . 5 |- ( v e. ( ~P J i^i ( LocFin ` J ) ) <-> ( v e. ~P J /\ v e. ( LocFin ` J ) ) )
23	22	anbi1i 731	. . . 4 |- ( ( v e. ( ~P J i^i ( LocFin ` J ) ) /\ v Ref U ) <-> ( ( v e. ~P J /\ v e. ( LocFin ` J ) ) /\ v Ref U ) )
24		anass 681	. . . 4 |- ( ( ( v e. ~P J /\ v e. ( LocFin ` J ) ) /\ v Ref U ) <-> ( v e. ~P J /\ ( v e. ( LocFin ` J ) /\ v Ref U ) ) )
25	23, 24	bitri 264	. . 3 |- ( ( v e. ( ~P J i^i ( LocFin ` J ) ) /\ v Ref U ) <-> ( v e. ~P J /\ ( v e. ( LocFin ` J ) /\ v Ref U ) ) )
26	25	rexbii2 3039	. 2 |- ( E. v e. ( ~P J i^i ( LocFin ` J ) ) v Ref U <-> E. v e. ~P J ( v e. ( LocFin ` J ) /\ v Ref U ) )
27	21, 26	sylib 208	1 |- ( ( J e. Paracomp /\ U C_ J /\ X = U. U ) -> E. v e. ~P J ( v e. ( LocFin ` J ) /\ v Ref U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   ` cfv 5888   Topctop 20698   Refcref 21305   LocFinclocfin 21307  CovHasRefccref 29909  Paracompcpcmp 29922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-cref 29910  df-pcmp 29923

This theorem is referenced by:  pcmplfinf  29928