Theorem reldv 25034

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 5607	. . . . . . . 8 ⊢ Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))
2	1	rgenw 3076	. . . . . . 7 ⊢ ∀𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))
3		reliun 5726	. . . . . . 7 ⊢ (Rel ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ ∀𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))
4	2, 3	mpbir 230	. . . . . 6 ⊢ Rel ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))
5		df-rel 5596	. . . . . 6 ⊢ (Rel ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ↔ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ⊆ (V × V))
6	4, 5	mpbi 229	. . . . 5 ⊢ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ⊆ (V × V)
7	6	rgenw 3076	. . . 4 ⊢ ∀𝑓 ∈ (ℂ ↑pm 𝑠)∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ⊆ (V × V)
8	7	rgenw 3076	. . 3 ⊢ ∀𝑠 ∈ 𝒫 ℂ∀𝑓 ∈ (ℂ ↑pm 𝑠)∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ⊆ (V × V)
9		df-dv 25031	. . . 4 ⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)))
10	9	ovmptss 7933	. . 3 ⊢ (∀𝑠 ∈ 𝒫 ℂ∀𝑓 ∈ (ℂ ↑pm 𝑠)∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ⊆ (V × V) → (𝑆 D 𝐹) ⊆ (V × V))
11	8, 10	ax-mp 5	. 2 ⊢ (𝑆 D 𝐹) ⊆ (V × V)
12		df-rel 5596	. 2 ⊢ (Rel (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) ⊆ (V × V))
13	11, 12	mpbir 230	1 ⊢ Rel (𝑆 D 𝐹)

Syntax hints:  ∀wral 3064  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561  ∪ ciun 4924   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275   ↑_pm cpm 8616  ℂcc 10869   − cmin 11205   / cdiv 11632   ↾_t crest 17131  TopOpenctopn 17132  ℂ_fldccnfld 20597  intcnt 22168   lim_ℂ climc 25026   D cdv 25027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-dv 25031

This theorem is referenced by:  perfdvf  25067  dvres  25075  dvres3  25077  dvres3a  25078  dvidlem  25079  dvmulbr  25103  dvaddf  25106  dvmulf  25107  dvcobr  25110  dvcof  25112  dvcnv  25141