MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldv Structured version   Visualization version   Unicode version

Theorem reldv 22874
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 4961 . . . . . . . 8  |-  Rel  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )
21rgenw 2761 . . . . . . 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 4973 . . . . . . 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 214 . . . . . 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 4860 . . . . . 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 213 . . . . 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 2761 . . . 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 2761 . . 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 22871 . . . 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 6904 . . 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 4860 . 2  |-  ( Rel  ( S  _D  F
)  <->  ( S  _D  F )  C_  ( _V  X.  _V ) )
1311, 12mpbir 214 1  |-  Rel  ( S  _D  F )
Colors of variables: wff setvar class
Syntax hints:   A.wral 2749   _Vcvv 3057    \ cdif 3413    C_ wss 3416   ~Pcpw 3963   {csn 3980   U_ciun 4292    |-> cmpt 4475    X. cxp 4851   dom cdm 4853   Rel wrel 4858   ` cfv 5601  (class class class)co 6315    ^pm cpm 7499   CCcc 9563    - cmin 9886    / cdiv 10297   ↾t crest 15368   TopOpenctopn 15369  ℂfldccnfld 19019   intcnt 20081   lim CC climc 22866    _D cdv 22867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-dv 22871
This theorem is referenced by:  perfdvf  22907  dvres  22915  dvres3  22917  dvres3a  22918  dvidlem  22919  dvmulbr  22942  dvaddf  22945  dvmulf  22946  dvcobr  22949  dvcof  22951  dvcnv  22978
  Copyright terms: Public domain W3C validator