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Theorem elcncf 11895
Description: Membership in the set of continuous complex functions from 
A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
Assertion
Ref Expression
elcncf  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  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 ) ) ) )
Distinct variable groups:    x, w, y, z, A    w, F, x, y, z    w, B, x, y, z

Proof of Theorem elcncf
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cncfval 11894 . . . 4  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  { 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 ) } )
21eleq2d 2158 . . 3  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  F  e.  { 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 ) } ) )
3 fveq1 5317 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
4 fveq1 5317 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  w )  =  ( F `  w ) )
53, 4oveq12d 5684 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  x
)  -  ( f `
 w ) )  =  ( ( F `
 x )  -  ( F `  w ) ) )
65fveq2d 5322 . . . . . . . 8  |-  ( f  =  F  ->  ( abs `  ( ( f `
 x )  -  ( f `  w
) ) )  =  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) ) )
76breq1d 3861 . . . . . . 7  |-  ( f  =  F  ->  (
( abs `  (
( f `  x
)  -  ( f `
 w ) ) )  <  y  <->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
87imbi2d 229 . . . . . 6  |-  ( f  =  F  ->  (
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( f `  x
)  -  ( f `
 w ) ) )  <  y )  <-> 
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
98rexralbidv 2405 . . . . 5  |-  ( f  =  F  ->  ( E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  < 
z  ->  ( abs `  ( ( f `  x )  -  (
f `  w )
) )  <  y
)  <->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
1092ralbidv 2403 . . . 4  |-  ( f  =  F  ->  ( 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 )  <->  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 ) ) )
1110elrab 2772 . . 3  |-  ( F  e.  { 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 ) }  <->  ( 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 ) ) )
122, 11syl6bb 195 . 2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( 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 ) ) ) )
13 cnex 7520 . . . . 5  |-  CC  e.  _V
1413ssex 3982 . . . 4  |-  ( B 
C_  CC  ->  B  e. 
_V )
1513ssex 3982 . . . 4  |-  ( A 
C_  CC  ->  A  e. 
_V )
16 elmapg 6432 . . . 4  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
1714, 15, 16syl2anr 285 . . 3  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( B  ^m  A )  <->  F : A
--> B ) )
1817anbi1d 454 . 2  |-  ( ( A  C_  CC  /\  B  C_  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 ) )  <->  ( F : A
--> B  /\  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 ) ) ) )
1912, 18bitrd 187 1  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   A.wral 2360   E.wrex 2361   {crab 2364   _Vcvv 2620    C_ wss 3000   class class class wbr 3851   -->wf 5024   ` cfv 5028  (class class class)co 5666    ^m cmap 6419   CCcc 7402    < clt 7576    - cmin 7707   RR+crp 9188   abscabs 10484   -cn->ccncf 11892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-map 6421  df-cncf 11893
This theorem is referenced by:  elcncf2  11896  cncff  11899  elcncf1di  11901  rescncf  11903
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