Theorem setsmstopn 21571

Description: The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)

Hypotheses

Ref	Expression
setsms.x	|- ( ph -> X = ( Base ` M ) )
setsms.d	|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k	|- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
setsms.m	|- ( ph -> M e. V )

Assertion

Ref	Expression
setsmstopn	|- ( ph -> ( MetOpen ` D ) = ( TopOpen ` K ) )

Proof of Theorem setsmstopn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setsms.x	. . 3 |- ( ph -> X = ( Base ` M ) )
2		setsms.d	. . 3 |- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
3		setsms.k	. . 3 |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
4		setsms.m	. . 3 |- ( ph -> M e. V )
5	1, 2, 3, 4	setsmstset 21570	. 2 |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )
6		df-mopn 19043	. . . . . . . 8 |- MetOpen = ( x e. U. ran *Met |-> ( topGen ` ran ( ball ` x ) ) )
7	6	dmmptss 5338	. . . . . . 7 |- dom MetOpen C_ U. ran *Met
8	7	sseli 3414	. . . . . 6 |- ( D e. dom MetOpen -> D e. U. ran *Met )
9		simpr 468	. . . . . . . . . . 11 |- ( ( ph /\ D e. U. ran *Met ) -> D e. U. ran *Met )
10		xmetunirn 21430	. . . . . . . . . . 11 |- ( D e. U. ran *Met <-> D e. ( *Met ` dom dom D ) )
11	9, 10	sylib 201	. . . . . . . . . 10 |- ( ( ph /\ D e. U. ran *Met ) -> D e. ( *Met ` dom dom D ) )
12		eqid 2471	. . . . . . . . . . 11 |- ( MetOpen ` D ) = ( MetOpen ` D )
13	12	mopnuni 21534	. . . . . . . . . 10 |- ( D e. ( *Met ` dom dom D ) -> dom dom D = U. ( MetOpen ` D ) )
14	11, 13	syl 17	. . . . . . . . 9 |- ( ( ph /\ D e. U. ran *Met ) -> dom dom D = U. ( MetOpen ` D ) )
15	2	dmeqd 5042	. . . . . . . . . . . . . 14 |- ( ph -> dom D = dom ( ( dist ` M ) |` ( X X. X ) ) )
16		dmres 5131	. . . . . . . . . . . . . 14 |- dom ( ( dist ` M ) |` ( X X. X ) ) = ( ( X X. X ) i^i dom ( dist ` M ) )
17	15, 16	syl6eq 2521	. . . . . . . . . . . . 13 |- ( ph -> dom D = ( ( X X. X ) i^i dom ( dist ` M ) ) )
18		inss1 3643	. . . . . . . . . . . . 13 |- ( ( X X. X ) i^i dom ( dist ` M ) ) C_ ( X X. X )
19	17, 18	syl6eqss 3468	. . . . . . . . . . . 12 |- ( ph -> dom D C_ ( X X. X ) )
20		dmss 5039	. . . . . . . . . . . 12 |- ( dom D C_ ( X X. X ) -> dom dom D C_ dom ( X X. X ) )
21	19, 20	syl 17	. . . . . . . . . . 11 |- ( ph -> dom dom D C_ dom ( X X. X ) )
22		dmxpid 5060	. . . . . . . . . . 11 |- dom ( X X. X ) = X
23	21, 22	syl6sseq 3464	. . . . . . . . . 10 |- ( ph -> dom dom D C_ X )
24	23	adantr 472	. . . . . . . . 9 |- ( ( ph /\ D e. U. ran *Met ) -> dom dom D C_ X )
25	14, 24	eqsstr3d 3453	. . . . . . . 8 |- ( ( ph /\ D e. U. ran *Met ) -> U. ( MetOpen ` D ) C_ X )
26		sspwuni 4360	. . . . . . . 8 |- ( ( MetOpen ` D ) C_ ~P X <-> U. ( MetOpen ` D ) C_ X )
27	25, 26	sylibr 217	. . . . . . 7 |- ( ( ph /\ D e. U. ran *Met ) -> ( MetOpen ` D ) C_ ~P X )
28	27	ex 441	. . . . . 6 |- ( ph -> ( D e. U. ran *Met -> ( MetOpen ` D ) C_ ~P X ) )
29	8, 28	syl5 32	. . . . 5 |- ( ph -> ( D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P X ) )
30		ndmfv 5903	. . . . . 6 |- ( -. D e. dom MetOpen -> ( MetOpen ` D ) = (/) )
31		0ss 3766	. . . . . 6 |- (/) C_ ~P X
32	30, 31	syl6eqss 3468	. . . . 5 |- ( -. D e. dom MetOpen -> ( MetOpen ` D ) C_ ~P X )
33	29, 32	pm2.61d1 164	. . . 4 |- ( ph -> ( MetOpen ` D ) C_ ~P X )
34	1, 2, 3	setsmsbas 21568	. . . . 5 |- ( ph -> X = ( Base ` K ) )
35	34	pweqd 3947	. . . 4 |- ( ph -> ~P X = ~P ( Base ` K ) )
36	33, 5, 35	3sstr3d 3460	. . 3 |- ( ph -> (TopSet ` K ) C_ ~P ( Base ` K ) )
37		eqid 2471	. . . 4 |- ( Base ` K ) = ( Base ` K )
38		eqid 2471	. . . 4 |- (TopSet ` K ) = (TopSet ` K )
39	37, 38	topnid 15412	. . 3 |- ( (TopSet ` K ) C_ ~P ( Base ` K ) -> (TopSet ` K ) = ( TopOpen ` K ) )
40	36, 39	syl 17	. 2 |- ( ph -> (TopSet ` K ) = ( TopOpen ` K ) )
41	5, 40	eqtrd 2505	1 |- ( ph -> ( MetOpen ` D ) = ( TopOpen ` K ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   <.cop 3965   U.cuni 4190   X. cxp 4837   dom cdm 4839   ran crn 4840   |` cres 4841   ` cfv 5589  (class class class)co 6308   ndxcnx 15196   sSet csts 15197   Basecbs 15199  TopSetcts 15274   distcds 15277   TopOpenctopn 15398   topGenctg 15414   *Metcxmt 19032   ballcbl 19034   MetOpencmopn 19037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-tset 15287  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000

This theorem is referenced by:  setsxms  21572  tmslem  21575