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Theorem metdsval 18367
Description: Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
Assertion
Ref Expression
metdsval  |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
Distinct variable groups:    x, y, A    x, D, y    x, S, y    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem metdsval
StepHypRef Expression
1 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x D y )  =  ( A D y ) )
21mpteq2dv 4123 . . . 4  |-  ( x  =  A  ->  (
y  e.  S  |->  ( x D y ) )  =  ( y  e.  S  |->  ( A D y ) ) )
32rneqd 4922 . . 3  |-  ( x  =  A  ->  ran  ( y  e.  S  |->  ( x D y ) )  =  ran  ( y  e.  S  |->  ( A D y ) ) )
43supeq1d 7215 . 2  |-  ( x  =  A  ->  sup ( ran  ( y  e.  S  |->  ( x D y ) ) , 
RR* ,  `'  <  )  =  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
5 metdscn.f . 2  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
6 xrltso 10491 . . . 4  |-  <  Or  RR*
7 cnvso 5230 . . . 4  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
86, 7mpbi 199 . . 3  |-  `'  <  Or 
RR*
98supex 7230 . 2  |-  sup ( ran  ( y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  )  e.  _V
104, 5, 9fvmpt 5618 1  |-  ( A  e.  X  ->  ( F `  A )  =  sup ( ran  (
y  e.  S  |->  ( A D y ) ) ,  RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093    Or wor 4329   `'ccnv 4704   ran crn 4706   ` cfv 5271  (class class class)co 5874   supcsup 7209   RR*cxr 8882    < clt 8883
This theorem is referenced by:  metdsge  18369  lebnumlem1  18475  lebnumlem3  18477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888
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