Theorem reldv 26199

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 5669	. . . . . . . 8 ⊢ Rel ({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))
2	1	rgenw 3080	. . . . . . 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 5795	. . . . . . 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 234	. . . . . 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 5658	. . . . . 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 233	. . . . 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 3080	. . . 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 3080	. . 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 26196	. . . 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 8108	. . 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 5658	. 2 ⊢ (Rel (𝑆 D 𝐹) ↔ (𝑆 D 𝐹) ⊆ (V × V))
13	11, 12	mpbir 234	1 ⊢ Rel (𝑆 D 𝐹)

This proof depends on syntax axioms:  ∀wral 3076  Vcvv 3450   ⊆ wss 3896   ∖ cdif 4044  𝒫 cpw 4556  {csn 4584  ∪ ciun 4950   ↦ cmpt 5185   × cxp 5649  dom cdm 5651  Rel wrel 5656  ‘cfv 6543  (class class class)co 7424   ↑_pm cpm 8855  ℂcc 11205   − cmin 11548   / cdiv 11980   ↾_t crest 17600  TopOpenctopn 17601  ℂ_fldccnfld 21687  intcnt 23344   lim_ℂ climc 26191   D cdv 26192

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-iota 6497  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-dv 26196

This theorem is used by:  perfdvf  26232  dvres  26240  dvres3  26242  dvres3a  26243  dvidlem  26244  dvmulbr  26268  dvaddf  26271  dvmulf  26272  dvcobr  26275  dvcof  26277  dvcnv  26306