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Theorem cncfrss2 14066
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 14061 . . 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 ) } )
21elmpocl2 6071 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
32elpwid 3587 1  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3130   ~Pcpw 3576   class class class wbr 4004   ` cfv 5217  (class class class)co 5875    ^m cmap 6648   CCcc 7809    < clt 7992    - cmin 8128   RR+crp 9653   abscabs 11006   -cn->ccncf 14060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-cncf 14061
This theorem is referenced by:  cncff  14067  cncfi  14068  rescncf  14071  climcncf  14074  cncfco  14081  cnlimci  14145
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