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Theorem reldv 24476
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 5560 . . . . . . . 8 Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
21rgenw 3145 . . . . . . 7 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
3 reliun 5676 . . . . . . 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 234 . . . . . 6 Rel 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥))
5 df-rel 5549 . . . . . 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 233 . . . . 5 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓𝑧) − (𝑓𝑥)) / (𝑧𝑥))) lim 𝑥)) ⊆ (V × V)
76rgenw 3145 . . . 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 3145 . . 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 24473 . . . 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 7784 . . 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 5549 . 2 (Rel (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) ⊆ (V × V))
1311, 12mpbir 234 1 Rel (𝑆 D 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wral 3133  Vcvv 3480  cdif 3916  wss 3919  𝒫 cpw 4522  {csn 4550   ciun 4905  cmpt 5132   × cxp 5540  dom cdm 5542  Rel wrel 5547  cfv 6343  (class class class)co 7149  pm cpm 8403  cc 10533  cmin 10868   / cdiv 11295  t crest 16694  TopOpenctopn 16695  fldccnfld 20545  intcnt 21625   lim climc 24468   D cdv 24469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-dv 24473
This theorem is referenced by:  perfdvf  24509  dvres  24517  dvres3  24519  dvres3a  24520  dvidlem  24521  dvmulbr  24545  dvaddf  24548  dvmulf  24549  dvcobr  24552  dvcof  24554  dvcnv  24583
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