Theorem dispcmp 29926

Description: Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
dispcmp	|- ( X e. V -> ~P X e. Paracomp )

Proof of Theorem dispcmp

Dummy variables v y z u x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 20799	. . 3 |- ( X e. V -> ~P X e. Top )
2		simpr 477	. . . . . . . . . . . 12 |- ( ( x e. X /\ u = { x } ) -> u = { x } )
3		snelpwi 4912	. . . . . . . . . . . . 13 |- ( x e. X -> { x } e. ~P X )
4	3	adantr 481	. . . . . . . . . . . 12 |- ( ( x e. X /\ u = { x } ) -> { x } e. ~P X )
5	2, 4	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( x e. X /\ u = { x } ) -> u e. ~P X )
6	5	rexlimiva 3028	. . . . . . . . . 10 |- ( E. x e. X u = { x } -> u e. ~P X )
7	6	abssi 3677	. . . . . . . . 9 |- { u | E. x e. X u = { x } } C_ ~P X
8		simpl 473	. . . . . . . . . . . . . 14 |- ( ( u = v /\ x = z ) -> u = v )
9		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( u = v /\ x = z ) -> x = z )
10	9	sneqd 4189	. . . . . . . . . . . . . 14 |- ( ( u = v /\ x = z ) -> { x } = { z } )
11	8, 10	eqeq12d 2637	. . . . . . . . . . . . 13 |- ( ( u = v /\ x = z ) -> ( u = { x } <-> v = { z } ) )
12	11	cbvrexdva 3178	. . . . . . . . . . . 12 |- ( u = v -> ( E. x e. X u = { x } <-> E. z e. X v = { z } ) )
13	12	cbvabv 2747	. . . . . . . . . . 11 |- { u | E. x e. X u = { x } } = { v | E. z e. X v = { z } }
14	13	dissnlocfin 21332	. . . . . . . . . 10 |- ( X e. V -> { u | E. x e. X u = { x } } e. ( LocFin ` ~P X ) )
15		elpwg 4166	. . . . . . . . . 10 |- ( { u | E. x e. X u = { x } } e. ( LocFin ` ~P X ) -> ( { u | E. x e. X u = { x } } e. ~P ~P X <-> { u | E. x e. X u = { x } } C_ ~P X ) )
16	14, 15	syl 17	. . . . . . . . 9 |- ( X e. V -> ( { u | E. x e. X u = { x } } e. ~P ~P X <-> { u | E. x e. X u = { x } } C_ ~P X ) )
17	7, 16	mpbiri 248	. . . . . . . 8 |- ( X e. V -> { u | E. x e. X u = { x } } e. ~P ~P X )
18	17	ad2antrr 762	. . . . . . 7 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> { u | E. x e. X u = { x } } e. ~P ~P X )
19	14	ad2antrr 762	. . . . . . 7 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> { u | E. x e. X u = { x } } e. ( LocFin ` ~P X ) )
20	18, 19	elind 3798	. . . . . 6 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> { u | E. x e. X u = { x } } e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) )
21		simpll 790	. . . . . . 7 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> X e. V )
22		simpr 477	. . . . . . . 8 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> X = U. y )
23	22	eqcomd 2628	. . . . . . 7 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> U. y = X )
24	13	dissnref 21331	. . . . . . 7 |- ( ( X e. V /\ U. y = X ) -> { u | E. x e. X u = { x } } Ref y )
25	21, 23, 24	syl2anc 693	. . . . . 6 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> { u | E. x e. X u = { x } } Ref y )
26		breq1 4656	. . . . . . 7 |- ( z = { u | E. x e. X u = { x } } -> ( z Ref y <-> { u | E. x e. X u = { x } } Ref y ) )
27	26	rspcev 3309	. . . . . 6 |- ( ( { u | E. x e. X u = { x } } e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) /\ { u | E. x e. X u = { x } } Ref y ) -> E. z e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) z Ref y )
28	20, 25, 27	syl2anc 693	. . . . 5 |- ( ( ( X e. V /\ y e. ~P ~P X ) /\ X = U. y ) -> E. z e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) z Ref y )
29	28	ex 450	. . . 4 |- ( ( X e. V /\ y e. ~P ~P X ) -> ( X = U. y -> E. z e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) z Ref y ) )
30	29	ralrimiva 2966	. . 3 |- ( X e. V -> A. y e. ~P ~P X ( X = U. y -> E. z e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) z Ref y ) )
31		unipw 4918	. . . . 5 |- U. ~P X = X
32	31	eqcomi 2631	. . . 4 |- X = U. ~P X
33	32	iscref 29911	. . 3 |- ( ~P X e. CovHasRef ( LocFin ` ~P X ) <-> ( ~P X e. Top /\ A. y e. ~P ~P X ( X = U. y -> E. z e. ( ~P ~P X i^i ( LocFin ` ~P X ) ) z Ref y ) ) )
34	1, 30, 33	sylanbrc 698	. 2 |- ( X e. V -> ~P X e. CovHasRef ( LocFin ` ~P X ) )
35		ispcmp 29924	. 2 |- ( ~P X e. Paracomp <-> ~P X e. CovHasRef ( LocFin ` ~P X ) )
36	34, 35	sylibr 224	1 |- ( X e. V -> ~P X e. Paracomp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {csn 4177   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-en 7956  df-fin 7959  df-top 20699  df-ref 21308  df-locfin 21310  df-cref 29910  df-pcmp 29923

This theorem is referenced by: (None)