Theorem dissnlocfin 21332

Description: The set of singletons is locally finite in the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.)

Hypothesis

Ref	Expression
dissnref.c	|- C = { u | E. x e. X u = { x } }

Assertion

Ref	Expression
dissnlocfin	|- ( X e. V -> C e. ( LocFin ` ~P X ) )

Distinct variable groups:   u, C, x   u, V, x   u, X, x

Proof of Theorem dissnlocfin

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

Step	Hyp	Ref	Expression
1		distop 20799	. 2 |- ( X e. V -> ~P X e. Top )
2		eqidd 2623	. 2 |- ( X e. V -> X = X )
3		snelpwi 4912	. . . . 5 |- ( z e. X -> { z } e. ~P X )
4	3	adantl 482	. . . 4 |- ( ( X e. V /\ z e. X ) -> { z } e. ~P X )
5		vsnid 4209	. . . . 5 |- z e. { z }
6	5	a1i 11	. . . 4 |- ( ( X e. V /\ z e. X ) -> z e. { z } )
7		nfv 1843	. . . . . 6 |- F/ u ( X e. V /\ z e. X )
8		nfrab1 3122	. . . . . 6 |- F/_ u { u e. C | ( u i^i { z } ) =/= (/) }
9		nfcv 2764	. . . . . 6 |- F/_ u { { z } }
10		dissnref.c	. . . . . . . . . 10 |- C = { u | E. x e. X u = { x } }
11	10	abeq2i 2735	. . . . . . . . 9 |- ( u e. C <-> E. x e. X u = { x } )
12	11	anbi1i 731	. . . . . . . 8 |- ( ( u e. C /\ ( u i^i { z } ) =/= (/) ) <-> ( E. x e. X u = { x } /\ ( u i^i { z } ) =/= (/) ) )
13		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) -> u = { x } )
14		simplr 792	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> u = { x } )
15	14	ineq1d 3813	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> ( u i^i { z } ) = ( { x } i^i { z } ) )
16		disjsn2 4247	. . . . . . . . . . . . . . . . . 18 |- ( x =/= z -> ( { x } i^i { z } ) = (/) )
17	16	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> ( { x } i^i { z } ) = (/) )
18	15, 17	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> ( u i^i { z } ) = (/) )
19		simp-4r 807	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> ( u i^i { z } ) =/= (/) )
20	19	neneqd 2799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) /\ x =/= z ) -> -. ( u i^i { z } ) = (/) )
21	18, 20	pm2.65da 600	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) -> -. x =/= z )
22		nne 2798	. . . . . . . . . . . . . . 15 |- ( -. x =/= z <-> x = z )
23	21, 22	sylib 208	. . . . . . . . . . . . . 14 |- ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) -> x = z )
24	23	sneqd 4189	. . . . . . . . . . . . 13 |- ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) -> { x } = { z } )
25	13, 24	eqtrd 2656	. . . . . . . . . . . 12 |- ( ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ x e. X ) /\ u = { x } ) -> u = { z } )
26	25	r19.29an 3077	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ z e. X ) /\ ( u i^i { z } ) =/= (/) ) /\ E. x e. X u = { x } ) -> u = { z } )
27	26	an32s 846	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ z e. X ) /\ E. x e. X u = { x } ) /\ ( u i^i { z } ) =/= (/) ) -> u = { z } )
28	27	anasss 679	. . . . . . . . 9 |- ( ( ( X e. V /\ z e. X ) /\ ( E. x e. X u = { x } /\ ( u i^i { z } ) =/= (/) ) ) -> u = { z } )
29		sneq 4187	. . . . . . . . . . . . 13 |- ( x = z -> { x } = { z } )
30	29	eqeq2d 2632	. . . . . . . . . . . 12 |- ( x = z -> ( u = { x } <-> u = { z } ) )
31	30	rspcev 3309	. . . . . . . . . . 11 |- ( ( z e. X /\ u = { z } ) -> E. x e. X u = { x } )
32	31	adantll 750	. . . . . . . . . 10 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> E. x e. X u = { x } )
33		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> u = { z } )
34	33	ineq1d 3813	. . . . . . . . . . . 12 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> ( u i^i { z } ) = ( { z } i^i { z } ) )
35		inidm 3822	. . . . . . . . . . . 12 |- ( { z } i^i { z } ) = { z }
36	34, 35	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> ( u i^i { z } ) = { z } )
37		vex 3203	. . . . . . . . . . . . 13 |- z e. _V
38	37	snnz 4309	. . . . . . . . . . . 12 |- { z } =/= (/)
39	38	a1i 11	. . . . . . . . . . 11 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> { z } =/= (/) )
40	36, 39	eqnetrd 2861	. . . . . . . . . 10 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> ( u i^i { z } ) =/= (/) )
41	32, 40	jca 554	. . . . . . . . 9 |- ( ( ( X e. V /\ z e. X ) /\ u = { z } ) -> ( E. x e. X u = { x } /\ ( u i^i { z } ) =/= (/) ) )
42	28, 41	impbida 877	. . . . . . . 8 |- ( ( X e. V /\ z e. X ) -> ( ( E. x e. X u = { x } /\ ( u i^i { z } ) =/= (/) ) <-> u = { z } ) )
43	12, 42	syl5bb 272	. . . . . . 7 |- ( ( X e. V /\ z e. X ) -> ( ( u e. C /\ ( u i^i { z } ) =/= (/) ) <-> u = { z } ) )
44		rabid 3116	. . . . . . 7 |- ( u e. { u e. C | ( u i^i { z } ) =/= (/) } <-> ( u e. C /\ ( u i^i { z } ) =/= (/) ) )
45		velsn 4193	. . . . . . 7 |- ( u e. { { z } } <-> u = { z } )
46	43, 44, 45	3bitr4g 303	. . . . . 6 |- ( ( X e. V /\ z e. X ) -> ( u e. { u e. C | ( u i^i { z } ) =/= (/) } <-> u e. { { z } } ) )
47	7, 8, 9, 46	eqrd 3622	. . . . 5 |- ( ( X e. V /\ z e. X ) -> { u e. C | ( u i^i { z } ) =/= (/) } = { { z } } )
48		snfi 8038	. . . . 5 |- { { z } } e. Fin
49	47, 48	syl6eqel 2709	. . . 4 |- ( ( X e. V /\ z e. X ) -> { u e. C | ( u i^i { z } ) =/= (/) } e. Fin )
50		eleq2 2690	. . . . . 6 |- ( y = { z } -> ( z e. y <-> z e. { z } ) )
51		ineq2 3808	. . . . . . . . 9 |- ( y = { z } -> ( u i^i y ) = ( u i^i { z } ) )
52	51	neeq1d 2853	. . . . . . . 8 |- ( y = { z } -> ( ( u i^i y ) =/= (/) <-> ( u i^i { z } ) =/= (/) ) )
53	52	rabbidv 3189	. . . . . . 7 |- ( y = { z } -> { u e. C | ( u i^i y ) =/= (/) } = { u e. C | ( u i^i { z } ) =/= (/) } )
54	53	eleq1d 2686	. . . . . 6 |- ( y = { z } -> ( { u e. C | ( u i^i y ) =/= (/) } e. Fin <-> { u e. C | ( u i^i { z } ) =/= (/) } e. Fin ) )
55	50, 54	anbi12d 747	. . . . 5 |- ( y = { z } -> ( ( z e. y /\ { u e. C | ( u i^i y ) =/= (/) } e. Fin ) <-> ( z e. { z } /\ { u e. C | ( u i^i { z } ) =/= (/) } e. Fin ) ) )
56	55	rspcev 3309	. . . 4 |- ( ( { z } e. ~P X /\ ( z e. { z } /\ { u e. C | ( u i^i { z } ) =/= (/) } e. Fin ) ) -> E. y e. ~P X ( z e. y /\ { u e. C | ( u i^i y ) =/= (/) } e. Fin ) )
57	4, 6, 49, 56	syl12anc 1324	. . 3 |- ( ( X e. V /\ z e. X ) -> E. y e. ~P X ( z e. y /\ { u e. C | ( u i^i y ) =/= (/) } e. Fin ) )
58	57	ralrimiva 2966	. 2 |- ( X e. V -> A. z e. X E. y e. ~P X ( z e. y /\ { u e. C | ( u i^i y ) =/= (/) } e. Fin ) )
59		unipw 4918	. . . 4 |- U. ~P X = X
60	59	eqcomi 2631	. . 3 |- X = U. ~P X
61	10	unisngl 21330	. . 3 |- X = U. C
62	60, 61	islocfin 21320	. 2 |- ( C e. ( LocFin ` ~P X ) <-> ( ~P X e. Top /\ X = X /\ A. z e. X E. y e. ~P X ( z e. y /\ { u e. C | ( u i^i y ) =/= (/) } e. Fin ) ) )
63	1, 2, 58, 62	syl3anbrc 1246	1 |- ( X e. V -> C e. ( LocFin ` ~P X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   ` cfv 5888   Fincfn 7955   Topctop 20698   LocFinclocfin 21307

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-locfin 21310

This theorem is referenced by:  dispcmp  29926