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Theorem elcncf1di 18399
Description: Membership in the set of continuous complex functions from 
A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
Hypotheses
Ref Expression
elcncf1d.1  |-  ( ph  ->  F : A --> B )
elcncf1d.2  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
elcncf1d.3  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
Assertion
Ref Expression
elcncf1di  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Distinct variable groups:    x, w, y, A    w, B, x, y    w, F, x, y    ph, w, x, y   
w, Z
Allowed substitution hints:    Z( x, y)

Proof of Theorem elcncf1di
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elcncf1d.1 . . 3  |-  ( ph  ->  F : A --> B )
2 elcncf1d.2 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
32imp 418 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  Z  e.  RR+ )
4 an32 773 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  <->  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) )
54anbi2i 675 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
6 anass 630 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
75, 6bitr4i 243 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ( ph  /\  ( x  e.  A  /\  y  e.  RR+ )
)  /\  w  e.  A ) )
8 elcncf1d.3 . . . . . . . 8  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
98imp 418 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
107, 9sylbir 204 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1110ralrimiva 2626 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
12 breq2 4027 . . . . . . . 8  |-  ( z  =  Z  ->  (
( abs `  (
x  -  w ) )  <  z  <->  ( abs `  ( x  -  w
) )  <  Z
) )
1312imbi1d 308 . . . . . . 7  |-  ( z  =  Z  ->  (
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <-> 
( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
1413ralbidv 2563 . . . . . 6  |-  ( z  =  Z  ->  ( A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <->  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
1514rspcev 2884 . . . . 5  |-  ( ( 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 ) )
163, 11, 15syl2anc 642 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  < 
z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1716ralrimivva 2635 . . 3  |-  ( ph  ->  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 ) )
181, 17jca 518 . 2  |-  ( ph  ->  ( 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 ) ) )
19 elcncf 18393 . 2  |-  ( ( 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 ) ) ) )
2018, 19syl5ibrcom 213 1  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719   -cn->ccncf 18380
This theorem is referenced by:  elcncf1ii  18400
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  ax-un 4512  ax-cnex 8793
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-sbc 2992  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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-cncf 18382
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