Theorem unirnblps 14590

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 14577	. . . 4 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5413	. . 3 |- ( D e. (PsMet ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 3997	. . 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 9723	. . . 4 |- 1 e. RR+
6		blcntrps 14583	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1337	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 9727	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrnps 14587	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1337	. . 3 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3840	. . 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 3198	1 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3153   ~Pcpw 3601   U.cuni 3835   X. cxp 4657   ran crn 4660   ` cfv 5254  (class class class)co 5918   1c1 7873   RR*cxr 8053   RR+crp 9719  PsMetcpsmet 14031   ballcbl 14034

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-0lt1 7978  ax-rnegex 7981

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-rp 9720  df-psmet 14039  df-bl 14042

This theorem is referenced by: (None)