Theorem dscopn 22378

Description: The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)

Hypothesis

Ref	Expression
dscmet.1	|- D = ( x e. X , y e. X |-> if ( x = y , 0 , 1 ) )

Assertion

Ref	Expression
dscopn	|- ( X e. V -> ( MetOpen ` D ) = ~P X )

Distinct variable group:   x, y, X

Allowed substitution hints:   D( x, y)   V( x, y)

Proof of Theorem dscopn

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dscmet.1	. . . . . . 7 |- D = ( x e. X , y e. X |-> if ( x = y , 0 , 1 ) )
2	1	dscmet 22377	. . . . . 6 |- ( X e. V -> D e. ( Met ` X ) )
3		metxmet 22139	. . . . . 6 |- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) )
4	2, 3	syl 17	. . . . 5 |- ( X e. V -> D e. ( *Met ` X ) )
5		eqid 2622	. . . . . 6 |- ( MetOpen ` D ) = ( MetOpen ` D )
6	5	elmopn 22247	. . . . 5 |- ( D e. ( *Met ` X ) -> ( u e. ( MetOpen ` D ) <-> ( u C_ X /\ A. v e. u E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) ) ) )
7	4, 6	syl 17	. . . 4 |- ( X e. V -> ( u e. ( MetOpen ` D ) <-> ( u C_ X /\ A. v e. u E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) ) ) )
8		simpll 790	. . . . . . . . 9 |- ( ( ( X e. V /\ u C_ X ) /\ v e. u ) -> X e. V )
9		ssel2 3598	. . . . . . . . . 10 |- ( ( u C_ X /\ v e. u ) -> v e. X )
10	9	adantll 750	. . . . . . . . 9 |- ( ( ( X e. V /\ u C_ X ) /\ v e. u ) -> v e. X )
11	8, 10	jca 554	. . . . . . . 8 |- ( ( ( X e. V /\ u C_ X ) /\ v e. u ) -> ( X e. V /\ v e. X ) )
12		velsn 4193	. . . . . . . . . . . 12 |- ( w e. { v } <-> w = v )
13		eleq1a 2696	. . . . . . . . . . . . . . 15 |- ( v e. X -> ( w = v -> w e. X ) )
14		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( w e. X /\ ( v D w ) < 1 ) -> w e. X )
15	14	a1i 11	. . . . . . . . . . . . . . 15 |- ( v e. X -> ( ( w e. X /\ ( v D w ) < 1 ) -> w e. X ) )
16		eqeq12 2635	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x = v /\ y = w ) -> ( x = y <-> v = w ) )
17	16	ifbid 4108	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = v /\ y = w ) -> if ( x = y , 0 , 1 ) = if ( v = w , 0 , 1 ) )
18		0re 10040	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. RR
19		1re 10039	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. RR
20	18, 19	keepel 4155	. . . . . . . . . . . . . . . . . . . . 21 |- if ( v = w , 0 , 1 ) e. RR
21	20	elexi 3213	. . . . . . . . . . . . . . . . . . . 20 |- if ( v = w , 0 , 1 ) e. _V
22	17, 1, 21	ovmpt2a 6791	. . . . . . . . . . . . . . . . . . 19 |- ( ( v e. X /\ w e. X ) -> ( v D w ) = if ( v = w , 0 , 1 ) )
23	22	breq1d 4663	. . . . . . . . . . . . . . . . . 18 |- ( ( v e. X /\ w e. X ) -> ( ( v D w ) < 1 <-> if ( v = w , 0 , 1 ) < 1 ) )
24	19	ltnri 10146	. . . . . . . . . . . . . . . . . . . . . 22 |- -. 1 < 1
25		iffalse 4095	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( -. v = w -> if ( v = w , 0 , 1 ) = 1 )
26	25	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . 22 |- ( -. v = w -> ( if ( v = w , 0 , 1 ) < 1 <-> 1 < 1 ) )
27	24, 26	mtbiri 317	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. v = w -> -. if ( v = w , 0 , 1 ) < 1 )
28	27	con4i 113	. . . . . . . . . . . . . . . . . . . 20 |- ( if ( v = w , 0 , 1 ) < 1 -> v = w )
29		iftrue 4092	. . . . . . . . . . . . . . . . . . . . 21 |- ( v = w -> if ( v = w , 0 , 1 ) = 0 )
30		0lt1 10550	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 1
31	29, 30	syl6eqbr 4692	. . . . . . . . . . . . . . . . . . . 20 |- ( v = w -> if ( v = w , 0 , 1 ) < 1 )
32	28, 31	impbii 199	. . . . . . . . . . . . . . . . . . 19 |- ( if ( v = w , 0 , 1 ) < 1 <-> v = w )
33		equcom 1945	. . . . . . . . . . . . . . . . . . 19 |- ( v = w <-> w = v )
34	32, 33	bitri 264	. . . . . . . . . . . . . . . . . 18 |- ( if ( v = w , 0 , 1 ) < 1 <-> w = v )
35	23, 34	syl6rbb 277	. . . . . . . . . . . . . . . . 17 |- ( ( v e. X /\ w e. X ) -> ( w = v <-> ( v D w ) < 1 ) )
36		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( v e. X /\ w e. X ) -> w e. X )
37	36	biantrurd 529	. . . . . . . . . . . . . . . . 17 |- ( ( v e. X /\ w e. X ) -> ( ( v D w ) < 1 <-> ( w e. X /\ ( v D w ) < 1 ) ) )
38	35, 37	bitrd 268	. . . . . . . . . . . . . . . 16 |- ( ( v e. X /\ w e. X ) -> ( w = v <-> ( w e. X /\ ( v D w ) < 1 ) ) )
39	38	ex 450	. . . . . . . . . . . . . . 15 |- ( v e. X -> ( w e. X -> ( w = v <-> ( w e. X /\ ( v D w ) < 1 ) ) ) )
40	13, 15, 39	pm5.21ndd 369	. . . . . . . . . . . . . 14 |- ( v e. X -> ( w = v <-> ( w e. X /\ ( v D w ) < 1 ) ) )
41	40	adantl 482	. . . . . . . . . . . . 13 |- ( ( X e. V /\ v e. X ) -> ( w = v <-> ( w e. X /\ ( v D w ) < 1 ) ) )
42		1rp 11836	. . . . . . . . . . . . . . . 16 |- 1 e. RR+
43		rpxr 11840	. . . . . . . . . . . . . . . 16 |- ( 1 e. RR+ -> 1 e. RR* )
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . 15 |- 1 e. RR*
45		elbl 22193	. . . . . . . . . . . . . . 15 |- ( ( D e. ( *Met ` X ) /\ v e. X /\ 1 e. RR* ) -> ( w e. ( v ( ball ` D ) 1 ) <-> ( w e. X /\ ( v D w ) < 1 ) ) )
46	44, 45	mp3an3 1413	. . . . . . . . . . . . . 14 |- ( ( D e. ( *Met ` X ) /\ v e. X ) -> ( w e. ( v ( ball ` D ) 1 ) <-> ( w e. X /\ ( v D w ) < 1 ) ) )
47	4, 46	sylan 488	. . . . . . . . . . . . 13 |- ( ( X e. V /\ v e. X ) -> ( w e. ( v ( ball ` D ) 1 ) <-> ( w e. X /\ ( v D w ) < 1 ) ) )
48	41, 47	bitr4d 271	. . . . . . . . . . . 12 |- ( ( X e. V /\ v e. X ) -> ( w = v <-> w e. ( v ( ball ` D ) 1 ) ) )
49	12, 48	syl5bb 272	. . . . . . . . . . 11 |- ( ( X e. V /\ v e. X ) -> ( w e. { v } <-> w e. ( v ( ball ` D ) 1 ) ) )
50	49	eqrdv 2620	. . . . . . . . . 10 |- ( ( X e. V /\ v e. X ) -> { v } = ( v ( ball ` D ) 1 ) )
51		blelrn 22222	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` X ) /\ v e. X /\ 1 e. RR* ) -> ( v ( ball ` D ) 1 ) e. ran ( ball ` D ) )
52	44, 51	mp3an3 1413	. . . . . . . . . . 11 |- ( ( D e. ( *Met ` X ) /\ v e. X ) -> ( v ( ball ` D ) 1 ) e. ran ( ball ` D ) )
53	4, 52	sylan 488	. . . . . . . . . 10 |- ( ( X e. V /\ v e. X ) -> ( v ( ball ` D ) 1 ) e. ran ( ball ` D ) )
54	50, 53	eqeltrd 2701	. . . . . . . . 9 |- ( ( X e. V /\ v e. X ) -> { v } e. ran ( ball ` D ) )
55		snssi 4339	. . . . . . . . . 10 |- ( v e. u -> { v } C_ u )
56		vsnid 4209	. . . . . . . . . 10 |- v e. { v }
57	55, 56	jctil 560	. . . . . . . . 9 |- ( v e. u -> ( v e. { v } /\ { v } C_ u ) )
58		eleq2 2690	. . . . . . . . . . 11 |- ( w = { v } -> ( v e. w <-> v e. { v } ) )
59		sseq1 3626	. . . . . . . . . . 11 |- ( w = { v } -> ( w C_ u <-> { v } C_ u ) )
60	58, 59	anbi12d 747	. . . . . . . . . 10 |- ( w = { v } -> ( ( v e. w /\ w C_ u ) <-> ( v e. { v } /\ { v } C_ u ) ) )
61	60	rspcev 3309	. . . . . . . . 9 |- ( ( { v } e. ran ( ball ` D ) /\ ( v e. { v } /\ { v } C_ u ) ) -> E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) )
62	54, 57, 61	syl2an 494	. . . . . . . 8 |- ( ( ( X e. V /\ v e. X ) /\ v e. u ) -> E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) )
63	11, 62	sylancom 701	. . . . . . 7 |- ( ( ( X e. V /\ u C_ X ) /\ v e. u ) -> E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) )
64	63	ralrimiva 2966	. . . . . 6 |- ( ( X e. V /\ u C_ X ) -> A. v e. u E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) )
65	64	ex 450	. . . . 5 |- ( X e. V -> ( u C_ X -> A. v e. u E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) ) )
66	65	pm4.71d 666	. . . 4 |- ( X e. V -> ( u C_ X <-> ( u C_ X /\ A. v e. u E. w e. ran ( ball ` D ) ( v e. w /\ w C_ u ) ) ) )
67	7, 66	bitr4d 271	. . 3 |- ( X e. V -> ( u e. ( MetOpen ` D ) <-> u C_ X ) )
68		selpw 4165	. . 3 |- ( u e. ~P X <-> u C_ X )
69	67, 68	syl6bbr 278	. 2 |- ( X e. V -> ( u e. ( MetOpen ` D ) <-> u e. ~P X ) )
70	69	eqrdv 2620	1 |- ( X e. V -> ( MetOpen ` D ) = ~P X )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   RR+crp 11832   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-pred 5680  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-bases 20750

This theorem is referenced by: (None)