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Theorem cncfrss 18921
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 18908 . . 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 6289 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 3808 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988    < clt 9120    - cmin 9291   RR+crp 10612   abscabs 12039   -cn->ccncf 18906
This theorem is referenced by:  cncff  18923  cncfi  18924  rescncf  18927  cncffvrn  18928  cncfco  18937  cncfmpt2f  18944  cncfcnvcn  18951  cncombf  19550  cnlimci  19776  ulmcn  20315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-cncf 18908
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