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Theorem blfps 20083
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 3521 . . . . . 6  |-  { y  e.  X  |  ( x D y )  <  r }  C_  X
2 elfvdm 5801 . . . . . . 7  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
3 elpw2g 4539 . . . . . . 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 2889 . . 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 2450 . . . 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 6727 . . 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 20060 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  =  ( x  e.  X , 
r  e.  RR*  |->  { y  e.  X  |  ( x D y )  <  r } ) )
1211feq1d 5630 . 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 369    e. wcel 1757   A.wral 2792   {crab 2796    C_ wss 3412   ~Pcpw 3944   class class class wbr 4376    X. cxp 4922   dom cdm 4924   -->wf 5498   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   RR*cxr 9504    < clt 9505  PsMetcpsmet 17895   ballcbl 17898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-map 7302  df-xr 9509  df-psmet 17904  df-bl 17907
This theorem is referenced by:  blrnps  20085  blelrnps  20093  unirnblps  20096
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