Theorem unirnblps 22224

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 22211	. . . 4 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2		frn 6053	. . . 4 |- ( ( ball ` D ) : ( X X. RR* ) --> ~P X -> ran ( ball ` D ) C_ ~P X )
3	1, 2	syl 17	. . 3 |- ( D e. (PsMet ` X ) -> ran ( ball ` D ) C_ ~P X )
4		sspwuni 4611	. . 3 |- ( ran ( ball ` D ) C_ ~P X <-> U. ran ( ball ` D ) C_ X )
5	3, 4	sylib 208	. 2 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) C_ X )
6		1rp 11836	. . . . . 6 |- 1 e. RR+
7		blcntrps 22217	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
8	6, 7	mp3an3 1413	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
9		rpxr 11840	. . . . . . 7 |- ( 1 e. RR+ -> 1 e. RR* )
10	6, 9	ax-mp 5	. . . . . 6 |- 1 e. RR*
11		blelrnps 22221	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12	10, 11	mp3an3 1413	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
13		elunii 4441	. . . . 5 |- ( ( x e. ( x ( ball ` D ) 1 ) /\ ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) ) -> x e. U. ran ( ball ` D ) )
14	8, 12, 13	syl2anc 693	. . . 4 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
15	14	ex 450	. . 3 |- ( D e. (PsMet ` X ) -> ( x e. X -> x e. U. ran ( ball ` D ) ) )
16	15	ssrdv 3609	. 2 |- ( D e. (PsMet ` X ) -> X C_ U. ran ( ball ` D ) )
17	5, 16	eqssd 3620	1 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1c1 9937   RR*cxr 10073   RR+crp 11832  PsMetcpsmet 19730   ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-rp 11833  df-psmet 19738  df-bl 19741

This theorem is referenced by:  psmetutop  22372