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Theorem cncfrss 20472
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 20459 . . 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 6310 . 2  |-  ( F  e.  ( A -cn-> B )  ->  A  e.  ~P CC )
32elpwid 3875 1  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724    C_ wss 3333   ~Pcpw 3865   class class class wbr 4297   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   CCcc 9285    < clt 9423    - cmin 9600   RR+crp 10996   abscabs 12728   -cn->ccncf 20457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-dm 4855  df-iota 5386  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-cncf 20459
This theorem is referenced by:  cncff  20474  cncfi  20475  rescncf  20478  cncffvrn  20479  cncfco  20488  cncfmpt2f  20495  cncfcnvcn  20502  cncombf  21141  cnlimci  21369  ulmcn  21869
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