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Theorem blf 20673
Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
blf  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )

Proof of Theorem blf
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3585 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
2 elfvdm 5892 . . . . . . 7  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
3 elpw2g 4610 . . . . . . 7  |-  ( X  e.  dom  *Met  ->  ( { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X 
<->  { y  e.  X  |  ( x D y )  <  r }  C_  X ) )
42, 3syl 16 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( { y  e.  X  |  ( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  | 
( x D y )  <  r } 
C_  X ) )
51, 4mpbiri 233 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X )
65a1d 25 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
( x  e.  X  /\  r  e.  RR* )  ->  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X
) )
76ralrimivv 2884 . . 3  |-  ( D  e.  ( *Met `  X )  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
8 eqid 2467 . . . 4  |-  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } )  =  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } )
98fmpt2 6851 . . 3  |-  ( A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X  <->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
107, 9sylib 196 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) : ( X  X.  RR* ) --> ~P X )
11 blfval 20650 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  =  ( x  e.  X ,  r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
1211feq1d 5717 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( ball `  D ) : ( X  X.  RR* ) --> ~P X  <->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X ) )
1310, 12mpbird 232 1  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   A.wral 2814   {crab 2818    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447    X. cxp 4997   dom cdm 4999   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   RR*cxr 9627    < clt 9628   *Metcxmt 18202   ballcbl 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-xr 9632  df-psmet 18210  df-xmet 18211  df-bl 18213
This theorem is referenced by:  blrn  20675  blelrn  20683  blssm  20684  unirnbl  20686  blin2  20695  imasf1oxms  20755  iscau2  21479  ismtyhmeolem  29931
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