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Theorem blfps 13569
Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blfps  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )

Proof of Theorem blfps
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3240 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
2 psmetrel 13482 . . . . . . . 8  |-  Rel PsMet
3 relelfvdm 5543 . . . . . . . 8  |-  ( ( Rel PsMet  /\  D  e.  (PsMet `  X ) )  ->  X  e.  dom PsMet )
42, 3mpan 424 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
5 elpw2g 4153 . . . . . . 7  |-  ( X  e.  dom PsMet  ->  ( { y  e.  X  | 
( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  |  (
x D y )  <  r }  C_  X ) )
64, 5syl 14 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( {
y  e.  X  | 
( x D y )  <  r }  e.  ~P X  <->  { y  e.  X  |  (
x D y )  <  r }  C_  X ) )
71, 6mpbiri 168 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  { y  e.  X  |  (
x D y )  <  r }  e.  ~P X )
87a1d 22 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( (
x  e.  X  /\  r  e.  RR* )  ->  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X
) )
98ralrimivv 2558 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
10 eqid 2177 . . . 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 } )
1110fmpo 6196 . . 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 )
129, 11sylib 122 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
13 blfvalps 13545 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
1413feq1d 5348 . 2  |-  ( D  e.  (PsMet `  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 ) )
1512, 14mpbird 167 1  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    e. wcel 2148   A.wral 2455   {crab 2459    C_ wss 3129   ~Pcpw 3574   class class class wbr 4000    X. cxp 4621   dom cdm 4623   Rel wrel 4628   -->wf 5208   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   RR*cxr 7978    < clt 7979  PsMetcpsmet 13139   ballcbl 13142
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890  ax-resscn 7891
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-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-pnf 7981  df-mnf 7982  df-xr 7983  df-psmet 13147  df-bl 13150
This theorem is referenced by:  blrnps  13571  blelrnps  13579  unirnblps  13582
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