Theorem unirnbl 14063

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 14050	. . . 4 |- ( D e. ( *Met ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2	1	frnd 5377	. . 3 |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ ~P X )
3		sspwuni 3973	. . 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 9660	. . . 4 |- 1 e. RR+
6		blcntr 14056	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR+ ) -> x e. ( x ( ball ` D ) 1 ) )
7	5, 6	mp3an3 1326	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> x e. ( x ( ball ` D ) 1 ) )
8		rpxr 9664	. . . . 5 |- ( 1 e. RR+ -> 1 e. RR* )
9	5, 8	ax-mp 5	. . . 4 |- 1 e. RR*
10		blelrn 14060	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. X /\ 1 e. RR* ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
11	9, 10	mp3an3 1326	. . 3 |- ( ( D e. ( *Met ` X ) /\ x e. X ) -> ( x ( ball ` D ) 1 ) e. ran ( ball ` D ) )
12		elunii 3816	. . 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. ( *Met ` X ) /\ x e. X ) -> x e. U. ran ( ball ` D ) )
14	4, 13	eqelssd 3176	1 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   X. cxp 4626   ran crn 4629   ` cfv 5218  (class class class)co 5878   1c1 7815   RR*cxr 7994   RR+crp 9656   *Metcxmet 13580   ballcbl 13582

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911  ax-0lt1 7920  ax-rnegex 7923

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-map 6653  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-rp 9657  df-psmet 13587  df-xmet 13588  df-bl 13590

This theorem is referenced by:  blbas  14073  mopntopon  14083  elmopn  14086  metss  14134  xmettx  14150