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Theorem reldv 21458
Description: The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
reldv  |-  Rel  ( S  _D  F )

Proof of Theorem reldv
Dummy variables  f 
s  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5042 . . . . . . . 8  |-  Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
21rgenw 2888 . . . . . . 7  |-  A. x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f ) Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
3 reliun 5055 . . . . . . 7  |-  ( Rel  U_ x  e.  (
( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  <->  A. x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f ) Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) ) )
42, 3mpbir 209 . . . . . 6  |-  Rel  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
5 df-rel 4942 . . . . . 6  |-  ( Rel  U_ x  e.  (
( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  <->  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  C_  ( _V  X.  _V ) )
64, 5mpbi 208 . . . . 5  |-  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  C_  ( _V  X.  _V )
76rgenw 2888 . . . 4  |-  A. f  e.  ( CC  ^pm  s
) U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  C_  ( _V  X.  _V )
87rgenw 2888 . . 3  |-  A. s  e.  ~P  CC A. f  e.  ( CC  ^pm  s
) U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  C_  ( _V  X.  _V )
9 df-dv 21455 . . . 4  |-  _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s ) 
|->  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) ) )
109ovmptss 6751 . . 3  |-  ( A. s  e.  ~P  CC A. f  e.  ( CC 
^pm  s ) U_ x  e.  ( ( int `  ( ( TopOpen ` fld )t  s
) ) `  dom  f ) ( { x }  X.  (
( z  e.  ( dom  f  \  {
x } )  |->  ( ( ( f `  z )  -  (
f `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( _V  X.  _V )  ->  ( S  _D  F )  C_  ( _V  X.  _V )
)
118, 10ax-mp 5 . 2  |-  ( S  _D  F )  C_  ( _V  X.  _V )
12 df-rel 4942 . 2  |-  ( Rel  ( S  _D  F
)  <->  ( S  _D  F )  C_  ( _V  X.  _V ) )
1311, 12mpbir 209 1  |-  Rel  ( S  _D  F )
Colors of variables: wff setvar class
Syntax hints:   A.wral 2793   _Vcvv 3065    \ cdif 3420    C_ wss 3423   ~Pcpw 3955   {csn 3972   U_ciun 4266    |-> cmpt 4445    X. cxp 4933   dom cdm 4935   Rel wrel 4940   ` cfv 5513  (class class class)co 6187    ^pm cpm 7312   CCcc 9378    - cmin 9693    / cdiv 10091   ↾t crest 14458   TopOpenctopn 14459  ℂfldccnfld 17924   intcnt 18734   lim CC climc 21450    _D cdv 21451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-dv 21455
This theorem is referenced by:  perfdvf  21491  dvres  21499  dvres3  21501  dvres3a  21502  dvidlem  21503  dvmulbr  21526  dvaddf  21529  dvmulf  21530  dvcobr  21533  dvcof  21535  dvcnv  21562
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