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Theorem cncfrss 21268
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 21255 . . 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 6503 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 4007 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1804   A.wral 2793   E.wrex 2794   {crab 2797    C_ wss 3461   ~Pcpw 3997   class class class wbr 4437   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   CCcc 9493    < clt 9631    - cmin 9810   RR+crp 11229   abscabs 13046   -cn->ccncf 21253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-dm 4999  df-iota 5541  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-cncf 21255
This theorem is referenced by:  cncff  21270  cncfi  21271  rescncf  21274  cncffvrn  21275  cncfco  21284  cncfmpt2f  21291  cncfcnvcn  21298  cncombf  21938  cnlimci  22166  ulmcn  22666  mulcncff  31577  subcncff  31589  negcncfg  31590  addcncff  31594  ioccncflimc  31595  icocncflimc  31599  divcncff  31601
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