Theorem unirnbl 15305

Description: The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Assertion

Ref	Expression
unirnbl	|- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )

Proof of Theorem unirnbl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		blf 15292	. . . 4 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5520	. . 3 |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 4078	. . 3 |- ( ran ( ball ` D ) C_ ~P X <-> U. ran ( ball ` D ) C_ X )
4	2, 3	sylib 122	. 2 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) C_ X )
5		1rp 9993	. . . 4 |- 1 e. RR+
6		blcntr 15298	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1363	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 9997	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrn 15302	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1363	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3921	. . 3 |- ( ( x e. ( x ( ball ` D ) 1 ) /\ ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) ) -> x e. U. ran ( ball ` D ) )
13	7, 11, 12	syl2anc 411	. 2 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
14	4, 13	eqelssd 3259	1 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3213   ~Pcpw 3671   U.cuni 3916   X. cxp 4749   ran crn 4752   ` cfv 5354  (class class class)co 6052   1c1 8130   RR*cxr 8309   RR+crp 9989   *Metcxmet 14701   ballcbl 14703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226  ax-0lt1 8235  ax-rnegex 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-rp 9990  df-psmet 14708  df-xmet 14709  df-bl 14711

This theorem is referenced by:  blbas  15315  mopntopon  15325  elmopn  15328  metss  15376  xmettx  15392