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Theorem cncffvrn 20479
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 20472 . . . 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 20471 . . 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 20475 . . . . . 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 1188 . . . . 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 2813 . . . 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 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285    < clt 9423    - cmin 9600   RR+crp 10996   abscabs 12728   -cn->ccncf 20457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385  df-cj 12593  df-re 12594  df-im 12595  df-abs 12730  df-cncf 20459
This theorem is referenced by:  cncfss  20480  cncfmpt2ss  20496  rolle  21467  dvlipcn  21471  c1lip2  21475  dvivthlem1  21485  dvivth  21487  lhop1lem  21490  dvcnvrelem2  21495  dvfsumlem2  21504  itgsubstlem  21525  efcvx  21919  dvrelog  22087  relogcn  22088  logcn  22097  dvlog  22101  logccv  22113  resqrcn  22192  loglesqr  22201  lgamgulmlem2  27021  areacirclem4  28492  cncfres  28669
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