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Theorem blfps 20994
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 3499 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
2 elfvdm 5800 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
3 elpw2g 4528 . . . . . . 7  |-  ( X  e.  dom PsMet  ->  ( { 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.  (PsMet `  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.  (PsMet `  X
)  ->  { y  e.  X  |  (
x D y )  <  r }  e.  ~P X )
65a1d 25 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( (
x  e.  X  /\  r  e.  RR* )  ->  { y  e.  X  |  ( x D y )  <  r }  e.  ~P X
) )
76ralrimivv 2802 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. x  e.  X  A. r  e.  RR*  { y  e.  X  |  ( x D y )  < 
r }  e.  ~P X )
8 eqid 2382 . . . 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 6766 . . 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.  (PsMet `  X
)  ->  ( x  e.  X ,  r  e. 
RR*  |->  { y  e.  X  |  ( x D y )  < 
r } ) : ( X  X.  RR* )
--> ~P X )
11 blfvalps 20971 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
1211feq1d 5625 . 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 ) )
1310, 12mpbird 232 1  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1826   A.wral 2732   {crab 2736    C_ wss 3389   ~Pcpw 3927   class class class wbr 4367    X. cxp 4911   dom cdm 4913   -->wf 5492   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   RR*cxr 9538    < clt 9539  PsMetcpsmet 18515   ballcbl 18518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-xr 9543  df-psmet 18524  df-bl 18527
This theorem is referenced by:  blrnps  20996  blelrnps  21004  unirnblps  21007
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