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Theorem blfps 12592
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 3182 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
2 psmetrel 12505 . . . . . . . 8  |-  Rel PsMet
3 relelfvdm 5453 . . . . . . . 8  |-  ( ( Rel PsMet  /\  D  e.  (PsMet `  X ) )  ->  X  e.  dom PsMet )
42, 3mpan 420 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
5 elpw2g 4081 . . . . . . 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 167 . . . . 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 2513 . . 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 2139 . . . 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 6099 . . 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 121 . 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 12568 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
1413feq1d 5259 . 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 166 1  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1480   A.wral 2416   {crab 2420    C_ wss 3071   ~Pcpw 3510   class class class wbr 3929    X. cxp 4537   dom cdm 4539   Rel wrel 4544   -->wf 5119   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   RR*cxr 7811    < clt 7812  PsMetcpsmet 12162   ballcbl 12165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-pnf 7814  df-mnf 7815  df-xr 7816  df-psmet 12170  df-bl 12173
This theorem is referenced by:  blrnps  12594  blelrnps  12602  unirnblps  12605
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