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Theorem cncfrss2 21131
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 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 ) } )
21elmpt2cl2 6501 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
32elpwid 4020 1  |-  ( F  e.  ( A -cn-> B )  ->  B  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  climcncf  21139  cncfco  21146  cncfcnvcn  21160  cnlimci  22028  cncfmptssg  31208  cncfcompt  31221
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