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Theorem reldv 19757
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 4983 . . . . . . . 8  |-  Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
21rgenw 2773 . . . . . . 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 4995 . . . . . . 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 201 . . . . . 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 4885 . . . . . 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 200 . . . . 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 2773 . . . 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 2773 . . 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 19754 . . . 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 6428 . . 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 8 . 2  |-  ( S  _D  F )  C_  ( _V  X.  _V )
12 df-rel 4885 . 2  |-  ( Rel  ( S  _D  F
)  <->  ( S  _D  F )  C_  ( _V  X.  _V ) )
1311, 12mpbir 201 1  |-  Rel  ( S  _D  F )
Colors of variables: wff set class
Syntax hints:   A.wral 2705   _Vcvv 2956    \ cdif 3317    C_ wss 3320   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266    X. cxp 4876   dom cdm 4878   Rel wrel 4883   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   CCcc 8988    - cmin 9291    / cdiv 9677   ↾t crest 13648   TopOpenctopn 13649  ℂfldccnfld 16703   intcnt 17081   lim CC climc 19749    _D cdv 19750
This theorem is referenced by:  perfdvf  19790  dvres  19798  dvres3  19800  dvres3a  19801  dvidlem  19802  dvmulbr  19825  dvaddf  19828  dvmulf  19829  dvcobr  19832  dvcof  19834  dvcnv  19861
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
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-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-dv 19754
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