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Theorem elcncf 21128
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 21127 . . . 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 2537 . . 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 5863 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
4 fveq1 5863 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  w )  =  ( F `  w ) )
53, 4oveq12d 6300 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f `  x
)  -  ( f `
 w ) )  =  ( ( F `
 x )  -  ( F `  w ) ) )
65fveq2d 5868 . . . . . . . 8  |-  ( f  =  F  ->  ( abs `  ( ( f `
 x )  -  ( f `  w
) ) )  =  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) ) )
76breq1d 4457 . . . . . . 7  |-  ( f  =  F  ->  (
( abs `  (
( f `  x
)  -  ( f `
 w ) ) )  <  y  <->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
87imbi2d 316 . . . . . 6  |-  ( f  =  F  ->  (
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( f `  x
)  -  ( f `
 w ) ) )  <  y )  <-> 
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
98rexralbidv 2981 . . . . 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 2908 . . . 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 3261 . . 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 261 . 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 9569 . . . . 5  |-  CC  e.  _V
1413ssex 4591 . . . 4  |-  ( B 
C_  CC  ->  B  e. 
_V )
1513ssex 4591 . . . 4  |-  ( A 
C_  CC  ->  A  e. 
_V )
16 elmapg 7430 . . . 4  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( F  e.  ( B  ^m  A )  <-> 
F : A --> B ) )
1714, 15, 16syl2anr 478 . . 3  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( B  ^m  A )  <->  F : A
--> B ) )
1817anbi1d 704 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486    < clt 9624    - cmin 9801   RR+crp 11216   abscabs 13026   -cn->ccncf 21115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-cncf 21117
This theorem is referenced by:  elcncf2  21129  cncff  21132  elcncf1di  21134  rescncf  21136  cncfmet  21147  cncfshift  31212  cncfperiod  31217
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