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Theorem reldv 25920
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.
StepHypRef Expression
1 relxp 5707 . . . . . . . 8 Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
21rgenw 3063 . . . . . . 7 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
3 reliun 5829 . . . . . . 7 (Rel 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ↔ ∀𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))
42, 3mpbir 231 . . . . . 6 Rel 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
5 df-rel 5696 . . . . . 6 (Rel 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ↔ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V))
64, 5mpbi 230 . . . . 5 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V)
76rgenw 3063 . . . 4 𝑓 ∈ (ℂ ↑pm 𝑠) 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V)
87rgenw 3063 . . 3 𝑠 ∈ 𝒫 ℂ∀𝑓 ∈ (ℂ ↑pm 𝑠) 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V)
9 df-dv 25917 . . . 4 D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)))
109ovmptss 8117 . . 3 (∀𝑠 ∈ 𝒫 ℂ∀𝑓 ∈ (ℂ ↑pm 𝑠) 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V) → (𝑆 D 𝐹) ⊆ (V × V))
118, 10ax-mp 5 . 2 (𝑆 D 𝐹) ⊆ (V × V)
12 df-rel 5696 . 2 (Rel (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) ⊆ (V × V))
1311, 12mpbir 231 1 Rel (𝑆 D 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wral 3059  Vcvv 3478  cdif 3960  wss 3963  𝒫 cpw 4605  {csn 4631   ciun 4996  cmpt 5231   × cxp 5687  dom cdm 5689  Rel wrel 5694  cfv 6563  (class class class)co 7431  pm cpm 8866  cc 11151  cmin 11490   / cdiv 11918  t crest 17467  TopOpenctopn 17468  fldccnfld 21382  intcnt 23041   lim climc 25912   D cdv 25913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-dv 25917
This theorem is referenced by:  perfdvf  25953  dvres  25961  dvres3  25963  dvres3a  25964  dvidlem  25965  dvmulbr  25990  dvmulbrOLD  25991  dvaddf  25994  dvmulf  25995  dvcobr  25998  dvcobrOLD  25999  dvcof  26001  dvcnv  26030
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