MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncfrss2 Unicode version

Theorem cncfrss2 18396
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 18382 . . 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 6063 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
3 elpwi 3633 . 2  |-  ( B  e.  ~P CC  ->  B 
C_  CC )
42, 3syl 15 1  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719   -cn->ccncf 18380
This theorem is referenced by:  cncff  18397  cncfi  18398  rescncf  18401  climcncf  18404  cncfco  18411  cncfcnvcn  18424  cnlimci  19239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-cncf 18382
  Copyright terms: Public domain W3C validator