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Theorem elcncf1di 18925
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 419 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  Z  e.  RR+ )
4 an32 774 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  <->  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) )
54anbi2i 676 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
6 anass 631 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
75, 6bitr4i 244 . . . . . . 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 419 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
107, 9sylbir 205 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1110ralrimiva 2789 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
12 breq2 4216 . . . . . . . 8  |-  ( z  =  Z  ->  (
( abs `  (
x  -  w ) )  <  z  <->  ( abs `  ( x  -  w
) )  <  Z
) )
1312imbi1d 309 . . . . . . 7  |-  ( z  =  Z  ->  (
( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y )  <-> 
( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
1413ralbidv 2725 . . . . . 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 3052 . . . . 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 643 . . . 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 2798 . . 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 519 . 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 18919 . 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 214 1  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    < clt 9120    - cmin 9291   RR+crp 10612   abscabs 12039   -cn->ccncf 18906
This theorem is referenced by:  elcncf1ii  18926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-cncf 18908
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