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

Theorem cncfrss2 20310
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 20296 . . 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 6295 . 2  |-  ( F  e.  ( A -cn-> B )  ->  B  e.  ~P CC )
32elpwid 3858 1  |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1755   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3316   ~Pcpw 3848   class class class wbr 4280   ` cfv 5406  (class class class)co 6080    ^m cmap 7202   CCcc 9268    < clt 9406    - cmin 9583   RR+crp 10979   abscabs 12707   -cn->ccncf 20294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-xp 4833  df-dm 4837  df-iota 5369  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-cncf 20296
This theorem is referenced by:  cncff  20311  cncfi  20312  rescncf  20315  climcncf  20318  cncfco  20325  cncfcnvcn  20339  cnlimci  21206
  Copyright terms: Public domain W3C validator