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Theorem reldv 19220
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 4794 . . . . . . . 8  |-  Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
21rgenw 2610 . . . . . . 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 4806 . . . . . . 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 200 . . . . . 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 4696 . . . . . 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 199 . . . . 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 2610 . . . 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 2610 . . 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 19217 . . . 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 6200 . . 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 4696 . 2  |-  ( Rel  ( S  _D  F
)  <->  ( S  _D  F )  C_  ( _V  X.  _V ) )
1311, 12mpbir 200 1  |-  Rel  ( S  _D  F )
Colors of variables: wff set class
Syntax hints:   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ~Pcpw 3625   {csn 3640   U_ciun 3905    e. cmpt 4077    X. cxp 4687   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735    - cmin 9037    / cdiv 9423   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   intcnt 16754   lim CC climc 19212    _D cdv 19213
This theorem is referenced by:  perfdvf  19253  dvres  19261  dvres3  19263  dvres3a  19264  dvidlem  19265  dvmulbr  19288  dvaddf  19291  dvmulf  19292  dvcobr  19295  dvcof  19297  dvcnv  19324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-dv 19217
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