Theorem reldv 25905

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 5703	. . . . . . . 8 ⊢ Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))
2	1	rgenw 3065	. . . . . . 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 5826	. . . . . . 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 231	. . . . . 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 5692	. . . . . 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 230	. . . . 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 3065	. . . 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 3065	. . 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 25902	. . . 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 8118	. . 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 5692	. 2 ⊢ (Rel (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) ⊆ (V × V))
13	11, 12	mpbir 231	1 ⊢ Rel (𝑆 D 𝐹)

Syntax hints:  ∀wral 3061  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ∪ ciun 4991   ↦ cmpt 5225   × cxp 5683  dom cdm 5685  Rel wrel 5690  ‘cfv 6561  (class class class)co 7431   ↑_pm cpm 8867  ℂcc 11153   − cmin 11492   / cdiv 11920   ↾_t crest 17465  TopOpenctopn 17466  ℂ_fldccnfld 21364  intcnt 23025   lim_ℂ climc 25897   D cdv 25898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-dv 25902

This theorem is referenced by:  perfdvf  25938  dvres  25946  dvres3  25948  dvres3a  25949  dvidlem  25950  dvmulbr  25975  dvmulbrOLD  25976  dvaddf  25979  dvmulf  25980  dvcobr  25983  dvcobrOLD  25984  dvcof  25986  dvcnv  26015