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Theorem cncfrss2 21373
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
Assertion
Ref Expression
cncfrss2  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )

Proof of Theorem cncfrss2
Dummy variables  a 
b  f  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cncf 21359 . . 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 ) } )
21elmpt2cl2 6504 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
32elpwid 4007 1  |-  ( F  e.  ( A -cn-> B )  ->  B  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 11230   abscabs 13048   -cn->ccncf 21357
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 21359
This theorem is referenced by:  cncff  21374  cncfi  21375  rescncf  21378  climcncf  21381  cncfco  21388  cncfcnvcn  21402  cnlimci  22270  cncfmptssg  31579  cncfcompt  31592
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