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

Theorem cncfrss2 18448
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 18434 . . 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 6105 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
3 elpwi 3667 . 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 1701   A.wral 2577   E.wrex 2578   {crab 2581    C_ wss 3186   ~Pcpw 3659   class class class wbr 4060   ` cfv 5292  (class class class)co 5900    ^m cmap 6815   CCcc 8780    < clt 8912    - cmin 9082   RR+crp 10401   abscabs 11766   -cn->ccncf 18432
This theorem is referenced by:  cncff  18449  cncfi  18450  rescncf  18453  climcncf  18456  cncfco  18463  cncfcnvcn  18477  cnlimci  19292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-dm 4736  df-iota 5256  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-cncf 18434
  Copyright terms: Public domain W3C validator