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Theorem cncffvrn 21137
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 21130 . . . 4  |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
21adantl 466 . . 3  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  A  C_  CC )
3 simpl 457 . . 3  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  C  C_  CC )
4 elcncf2 21129 . . 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 661 . 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 21133 . . . . . 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 1197 . . . . 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 2885 . . . 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 466 . . 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 507 . 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 256 1  |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) )  ->  ( F  e.  ( A -cn-> C )  <-> 
F : A --> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447   -->wf 5582   ` cfv 5586  (class class class)co 6282   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  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  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-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-cj 12891  df-re 12892  df-im 12893  df-abs 13028  df-cncf 21117
This theorem is referenced by:  cncfss  21138  cncfmpt2ss  21154  rolle  22126  dvlipcn  22130  c1lip2  22134  dvivthlem1  22144  dvivth  22146  lhop1lem  22149  dvcnvrelem2  22154  dvfsumlem2  22163  itgsubstlem  22184  efcvx  22578  dvrelog  22746  relogcn  22747  logcn  22756  dvlog  22760  logccv  22772  resqrtcn  22851  loglesqrt  22860  lgamgulmlem2  28212  areacirclem4  29687  cncfres  29864  cncfmptssg  31208  resincncf  31213  cncfcompt  31221  cncfiooiccre  31234  dvdivcncf  31257  dvbdfbdioolem1  31258  ioodvbdlimc1lem2  31262  ioodvbdlimc2lem  31264  itgsbtaddcnst  31300  fourierdlem58  31465  fourierdlem59  31466  fourierdlem62  31469  fourierdlem68  31475  fourierdlem76  31483  fourierdlem78  31485  fourierdlem83  31490  fourierdlem101  31508  fourierdlem112  31519  fouriercn  31533
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