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Theorem reldv 22440
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 5098 . . . . . . . 8  |-  Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
21rgenw 2815 . . . . . . 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 5111 . . . . . . 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 4995 . . . . . 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 2815 . . . 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 2815 . . 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 22437 . . . 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 6854 . . 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 4995 . 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 2804   _Vcvv 3106    \ cdif 3458    C_ wss 3461   ~Pcpw 3999   {csn 4016   U_ciun 4315    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   Rel wrel 4993   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   CCcc 9479    - cmin 9796    / cdiv 10202   ↾t crest 14910   TopOpenctopn 14911  ℂfldccnfld 18615   intcnt 19685   lim CC climc 22432    _D cdv 22433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-dv 22437
This theorem is referenced by:  perfdvf  22473  dvres  22481  dvres3  22483  dvres3a  22484  dvidlem  22485  dvmulbr  22508  dvaddf  22511  dvmulf  22512  dvcobr  22515  dvcof  22517  dvcnv  22544
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