Theorem locfindis 21333

Description: The locally finite covers of a discrete space are precisely the point-finite covers. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
locfindis.1	|- Y = U. C

Assertion

Ref	Expression
locfindis	|- ( C e. ( LocFin ` ~P X ) <-> ( C e. PtFin /\ X = Y ) )

Proof of Theorem locfindis

Dummy variables x s y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lfinpfin 21327	. . 3 |- ( C e. ( LocFin ` ~P X ) -> C e. PtFin )
2		unipw 4918	. . . . 5 |- U. ~P X = X
3	2	eqcomi 2631	. . . 4 |- X = U. ~P X
4		locfindis.1	. . . 4 |- Y = U. C
5	3, 4	locfinbas 21325	. . 3 |- ( C e. ( LocFin ` ~P X ) -> X = Y )
6	1, 5	jca 554	. 2 |- ( C e. ( LocFin ` ~P X ) -> ( C e. PtFin /\ X = Y ) )
7		simpr 477	. . . . 5 |- ( ( C e. PtFin /\ X = Y ) -> X = Y )
8		uniexg 6955	. . . . . . 7 |- ( C e. PtFin -> U. C e. _V )
9	4, 8	syl5eqel 2705	. . . . . 6 |- ( C e. PtFin -> Y e. _V )
10	9	adantr 481	. . . . 5 |- ( ( C e. PtFin /\ X = Y ) -> Y e. _V )
11	7, 10	eqeltrd 2701	. . . 4 |- ( ( C e. PtFin /\ X = Y ) -> X e. _V )
12		distop 20799	. . . 4 |- ( X e. _V -> ~P X e. Top )
13	11, 12	syl 17	. . 3 |- ( ( C e. PtFin /\ X = Y ) -> ~P X e. Top )
14		snelpwi 4912	. . . . . 6 |- ( x e. X -> { x } e. ~P X )
15	14	adantl 482	. . . . 5 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> { x } e. ~P X )
16		snidg 4206	. . . . . 6 |- ( x e. X -> x e. { x } )
17	16	adantl 482	. . . . 5 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> x e. { x } )
18		simpll 790	. . . . . 6 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> C e. PtFin )
19	7	eleq2d 2687	. . . . . . 7 |- ( ( C e. PtFin /\ X = Y ) -> ( x e. X <-> x e. Y ) )
20	19	biimpa 501	. . . . . 6 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> x e. Y )
21	4	ptfinfin 21322	. . . . . 6 |- ( ( C e. PtFin /\ x e. Y ) -> { s e. C | x e. s } e. Fin )
22	18, 20, 21	syl2anc 693	. . . . 5 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> { s e. C | x e. s } e. Fin )
23		eleq2 2690	. . . . . . 7 |- ( y = { x } -> ( x e. y <-> x e. { x } ) )
24		ineq2 3808	. . . . . . . . . . 11 |- ( y = { x } -> ( s i^i y ) = ( s i^i { x } ) )
25	24	neeq1d 2853	. . . . . . . . . 10 |- ( y = { x } -> ( ( s i^i y ) =/= (/) <-> ( s i^i { x } ) =/= (/) ) )
26		disjsn 4246	. . . . . . . . . . 11 |- ( ( s i^i { x } ) = (/) <-> -. x e. s )
27	26	necon2abii 2844	. . . . . . . . . 10 |- ( x e. s <-> ( s i^i { x } ) =/= (/) )
28	25, 27	syl6bbr 278	. . . . . . . . 9 |- ( y = { x } -> ( ( s i^i y ) =/= (/) <-> x e. s ) )
29	28	rabbidv 3189	. . . . . . . 8 |- ( y = { x } -> { s e. C | ( s i^i y ) =/= (/) } = { s e. C | x e. s } )
30	29	eleq1d 2686	. . . . . . 7 |- ( y = { x } -> ( { s e. C | ( s i^i y ) =/= (/) } e. Fin <-> { s e. C | x e. s } e. Fin ) )
31	23, 30	anbi12d 747	. . . . . 6 |- ( y = { x } -> ( ( x e. y /\ { s e. C | ( s i^i y ) =/= (/) } e. Fin ) <-> ( x e. { x } /\ { s e. C | x e. s } e. Fin ) ) )
32	31	rspcev 3309	. . . . 5 |- ( ( { x } e. ~P X /\ ( x e. { x } /\ { s e. C | x e. s } e. Fin ) ) -> E. y e. ~P X ( x e. y /\ { s e. C | ( s i^i y ) =/= (/) } e. Fin ) )
33	15, 17, 22, 32	syl12anc 1324	. . . 4 |- ( ( ( C e. PtFin /\ X = Y ) /\ x e. X ) -> E. y e. ~P X ( x e. y /\ { s e. C | ( s i^i y ) =/= (/) } e. Fin ) )
34	33	ralrimiva 2966	. . 3 |- ( ( C e. PtFin /\ X = Y ) -> A. x e. X E. y e. ~P X ( x e. y /\ { s e. C | ( s i^i y ) =/= (/) } e. Fin ) )
35	3, 4	islocfin 21320	. . 3 |- ( C e. ( LocFin ` ~P X ) <-> ( ~P X e. Top /\ X = Y /\ A. x e. X E. y e. ~P X ( x e. y /\ { s e. C | ( s i^i y ) =/= (/) } e. Fin ) ) )
36	13, 7, 34, 35	syl3anbrc 1246	. 2 |- ( ( C e. PtFin /\ X = Y ) -> C e. ( LocFin ` ~P X ) )
37	6, 36	impbii 199	1 |- ( C e. ( LocFin ` ~P X ) <-> ( C e. PtFin /\ X = Y ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   ` cfv 5888   Fincfn 7955   Topctop 20698   PtFincptfin 21306   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-er 7742  df-en 7956  df-fin 7959  df-top 20699  df-ptfin 21309  df-locfin 21310

This theorem is referenced by: (None)