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.

Step	Hyp	Ref	Expression
1		relxp 5098	. . . . . . . 8 |- Rel ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) )
2	1	rgenw 2815	. . . . . . 7 |- A. x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾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 ` ℂfld ) ↾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 ` ℂfld ) ↾t s ) ) ` dom f ) Rel ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
4	2, 3	mpbir 209	. . . . . 6 |- Rel U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾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 ` ℂfld ) ↾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 ` ℂfld ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( _V X. _V ) )
6	4, 5	mpbi 208	. . . . 5 |- U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( _V X. _V )
7	6	rgenw 2815	. . . 4 |- A. f e. ( CC ^pm s ) U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) C_ ( _V X. _V )
8	7	rgenw 2815	. . 3 |- A. s e. ~P CC A. f e. ( CC ^pm s ) U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾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 ` ℂfld ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
10	9	ovmptss 6854	. . 3 |- ( A. s e. ~P CC A. f e. ( CC ^pm s ) U_ x e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾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 ) )
11	8, 10	ax-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 ) )
13	11, 12	mpbir 209	1 |- Rel ( S _D F )

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  ℂ_fldccnfld 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