Theorem unirnblps 15446

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.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
unirnblps	|- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )

Proof of Theorem unirnblps

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		blfps 15433	. . . 4 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5538	. . 3 |- ( D e. (PsMet ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 4092	. . 3 |- ( ran ( ball ` D ) C_ ~P X <-> U. ran ( ball ` D ) C_ X )
4	2, 3	sylib 122	. 2 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) C_ X )
5		1rp 10037	. . . 4 |- 1 e. RR+
6		blcntrps 15439	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1367	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 10041	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrnps 15443	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1367	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3935	. . 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. (PsMet ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
14	4, 13	eqelssd 3267	1 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   X. cxp 4767   ran crn 4770   ` cfv 5372  (class class class)co 6075   1c1 8170   RR*cxr 8349   RR+crp 10033  PsMetcpsmet 14844   ballcbl 14847

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-0lt1 8275  ax-rnegex 8278

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-rp 10034  df-psmet 14852  df-bl 14855

This theorem is referenced by: (None)