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Theorem cncfrss 12343
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
Assertion
Ref Expression
cncfrss  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )

Proof of Theorem cncfrss
Dummy variables  a  b  f  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cncf 12339 . . 3  |-  -cn->  =  ( a  e.  ~P CC ,  b  e.  ~P CC  |->  { f  e.  ( b  ^m  a
)  |  A. x  e.  a  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  a  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( f `  x
)  -  ( f `
 w ) ) )  <  y ) } )
21elmpt2cl1 5881 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 3460 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1445   A.wral 2370   E.wrex 2371   {crab 2374    C_ wss 3013   ~Pcpw 3449   class class class wbr 3867   ` cfv 5049  (class class class)co 5690    ^m cmap 6445   CCcc 7445    < clt 7619    - cmin 7750   RR+crp 9233   abscabs 10561   -cn->ccncf 12338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-cncf 12339
This theorem is referenced by:  cncff  12345  cncfi  12346  rescncf  12349  cncffvrn  12350  cncfco  12359
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