Theorem reldv 23634

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 5227	. . . . . . . 8 |- Rel ( { x } X. ( ( z e. ( dom f \ { x } ) |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) )
2	1	rgenw 2924	. . . . . . 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 5239	. . . . . . 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 221	. . . . . 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 5121	. . . . . 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 220	. . . . 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 2924	. . . 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 2924	. . 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 23631	. . . 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 7258	. . 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 5121	. 2 |- ( Rel ( S _D F ) <-> ( S _D F ) C_ ( _V X. _V ) )
13	11, 12	mpbir 221	1 |- Rel ( S _D F )

Syntax hints:   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   - cmin 10266   / cdiv 10684   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   intcnt 20821   lim CC climc 23626   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-dv 23631

This theorem is referenced by:  perfdvf  23667  dvres  23675  dvres3  23677  dvres3a  23678  dvidlem  23679  dvmulbr  23702  dvaddf  23705  dvmulf  23706  dvcobr  23709  dvcof  23711  dvcnv  23740