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Theorem cncffvrn 18800
Description: Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
cncffvrn  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  ( F  e.  ( A -cn-> C )  <-> 
F : A --> C ) )

Proof of Theorem cncffvrn
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cncfrss 18793 . . . 4  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
21adantl 453 . . 3  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  A  C_  CC )
3 simpl 444 . . 3  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  C  C_  CC )
4 elcncf2 18792 . . 3  |-  ( ( A  C_  CC  /\  C  C_  CC )  ->  ( F  e.  ( A -cn-> C )  <->  ( F : A --> C  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
52, 3, 4syl2anc 643 . 2  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  ( F  e.  ( A -cn-> C )  <-> 
( F : A --> C  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
6 cncfi 18796 . . . . . 6  |-  ( ( F  e.  ( A
-cn-> B )  /\  x  e.  A  /\  y  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) )
763expb 1154 . . . . 5  |-  ( ( F  e.  ( A
-cn-> B )  /\  (
x  e.  A  /\  y  e.  RR+ ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) )
87ralrimivva 2742 . . . 4  |-  ( F  e.  ( A -cn-> B )  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) )
98adantl 453 . . 3  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
w  -  x ) )  <  z  -> 
( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) )
109biantrud 494 . 2  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  ( F : A --> C  <->  ( F : A --> C  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x
) )  <  z  ->  ( abs `  (
( F `  w
)  -  ( F `
 x ) ) )  <  y ) ) ) )
115, 10bitr4d 248 1  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  ( F  e.  ( A -cn-> C )  <-> 
F : A --> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   A.wral 2650   E.wrex 2651    C_ wss 3264   class class class wbr 4154   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922    < clt 9054    - cmin 9224   RR+crp 10545   abscabs 11967   -cn->ccncf 18778
This theorem is referenced by:  cncfss  18801  cncfmpt2ss  18817  rolle  19742  dvlipcn  19746  c1lip2  19750  dvivthlem1  19760  dvivth  19762  lhop1lem  19765  dvcnvrelem2  19770  dvfsumlem2  19779  itgsubstlem  19800  efcvx  20233  dvrelog  20396  relogcn  20397  logcn  20406  dvlog  20410  logccv  20422  resqrcn  20501  loglesqr  20510  lgamgulmlem2  24594  areacirclem5  25987  cncfres  26166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-2 9991  df-cj 11832  df-re 11833  df-im 11834  df-abs 11969  df-cncf 18780
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