Theorem unirnblps 15413

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 15400	. . . 4 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5523	. . 3 |- ( D e. (PsMet ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 4081	. . 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 10008	. . . 4 |- 1 e. RR+
6		blcntrps 15406	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1363	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 10012	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrnps 15410	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1363	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3924	. . 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. (PsMet ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
14	4, 13	eqelssd 3261	1 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   C_ wss 3214   ~Pcpw 3674   U.cuni 3919   X. cxp 4752   ran crn 4755   ` cfv 5357  (class class class)co 6058   1c1 8144   RR*cxr 8323   RR+crp 10004  PsMetcpsmet 14809   ballcbl 14812

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240  ax-0lt1 8249  ax-rnegex 8252

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-rp 10005  df-psmet 14817  df-bl 14820

This theorem is referenced by: (None)