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.

Step	Hyp	Ref	Expression
1		relxp 4961	. . . . . . . 8 |- Rel ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) )
2	1	rgenw 2761	. . . . . . 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 4973	. . . . . . 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 214	. . . . . 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 4860	. . . . . 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 213	. . . . 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 2761	. . . 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 2761	. . 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 22871	. . . 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 6904	. . 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 4860	. 2 |- ( Rel ( S _D F ) <-> ( S _D F ) C_ ( _V X. _V ) )
13	11, 12	mpbir 214	1 |- Rel ( S _D F )

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  ℂ_fldccnfld 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