Theorem unirnbl 15450

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 15437	. . . 4 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5541	. . 3 |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 4095	. . 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 10040	. . . 4 |- 1 e. RR+
6		blcntr 15443	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1367	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 10044	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrn 15447	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1367	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3938	. . 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 415	. 2 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
14	4, 13	eqelssd 3267	1 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3688   U.cuni 3933   X. cxp 4770   ran crn 4773   ` cfv 5375  (class class class)co 6078   1c1 8173   RR*cxr 8352   RR+crp 10036   *Metcxmet 14848   ballcbl 14850

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1re 8266  ax-addrcl 8269  ax-0lt1 8278  ax-rnegex 8281

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-map 6917  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-rp 10037  df-psmet 14855  df-xmet 14856  df-bl 14858

This theorem is referenced by:  blbas  15460  mopntopon  15470  elmopn  15473  metss  15521  xmettx  15537