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Theorem cncfrss 21689
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 21676 . . 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 6501 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 3967 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1844   A.wral 2756   E.wrex 2757   {crab 2760    C_ wss 3416   ~Pcpw 3957   class class class wbr 4397   ` cfv 5571  (class class class)co 6280    ^m cmap 7459   CCcc 9522    < clt 9660    - cmin 9843   RR+crp 11267   abscabs 13218   -cn->ccncf 21674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-xp 4831  df-dm 4835  df-iota 5535  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-cncf 21676
This theorem is referenced by:  cncff  21691  cncfi  21692  rescncf  21695  cncffvrn  21696  cncfco  21705  cncfmpt2f  21712  cncfcnvcn  21719  cncombf  22359  cnlimci  22587  ulmcn  23088  mulcncff  37051  subcncff  37063  negcncfg  37064  addcncff  37068  ioccncflimc  37069  icocncflimc  37073  divcncff  37075  cncfcompt2  37083
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