Theorem reldv 21458

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 5042	. . . . . . . 8 |- Rel ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) )
2	1	rgenw 2888	. . . . . . 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 5055	. . . . . . 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 4942	. . . . . 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 2888	. . . 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 2888	. . 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 21455	. . . 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 6751	. . 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 4942	. 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 2793   _Vcvv 3065   \ cdif 3420   C_ wss 3423   ~Pcpw 3955   {csn 3972   U_ciun 4266   |-> cmpt 4445   X. cxp 4933   dom cdm 4935   Rel wrel 4940   ` cfv 5513  (class class class)co 6187   ^pm cpm 7312   CCcc 9378   - cmin 9693   / cdiv 10091   ↾_t crest 14458   TopOpenctopn 14459  ℂ_fldccnfld 17924   intcnt 18734   lim CC climc 21450   _D cdv 21451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-dv 21455

This theorem is referenced by:  perfdvf  21491  dvres  21499  dvres3  21501  dvres3a  21502  dvidlem  21503  dvmulbr  21526  dvaddf  21529  dvmulf  21530  dvcobr  21533  dvcof  21535  dvcnv  21562