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Theorem cncfrss 21130
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 21117 . . 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 6500 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 4020 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486    < clt 9624    - cmin 9801   RR+crp 11216   abscabs 13026   -cn->ccncf 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-dm 5009  df-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-cncf 21117
This theorem is referenced by:  cncff  21132  cncfi  21133  rescncf  21136  cncffvrn  21137  cncfco  21146  cncfmpt2f  21153  cncfcnvcn  21160  cncombf  21800  cnlimci  22028  ulmcn  22528  mulcncff  31206  subcncff  31218  negcncfg  31219  addcncff  31223  ioccncflimc  31224  icocncflimc  31228  divcncff  31230
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