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Theorem dvbsss 25879

Description: The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.)

**Assertion**
Ref	Expression
dvbsss	⊢ dom (𝑆 D 𝐹) ⊆ 𝑆

Proof of Theorem dvbsss Dummy variables 𝑓 𝑠 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dv 25844	. . . . . . . . . . 11 ⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑_pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
2	1	reldmmpo 7494	. . . . . . . . . 10 ⊢ Rel dom D
3		df-rel 5631	. . . . . . . . . 10 ⊢ (Rel dom D ↔ dom D ⊆ (V × V))
4	2, 3	mpbi 230	. . . . . . . . 9 ⊢ dom D ⊆ (V × V)
5	4	sseli 3918	. . . . . . . 8 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → ⟨𝑆, 𝐹⟩ ∈ (V × V))
6		opelxp1 5666	. . . . . . . 8 ⊢ (⟨𝑆, 𝐹⟩ ∈ (V × V) → 𝑆 ∈ V)
7	5, 6	syl 17	. . . . . . 7 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ V)
8		opeq1 4817	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ⟨𝑠, 𝐹⟩ = ⟨𝑆, 𝐹⟩)
9	8	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (⟨𝑠, 𝐹⟩ ∈ dom D ↔ ⟨𝑆, 𝐹⟩ ∈ dom D ))
10		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 ℂ ↔ 𝑆 ∈ 𝒫 ℂ))
11		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
12	11	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (ℂ ↑_pm 𝑠) ↔ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
13	10, 12	anbi12d 633	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑠)) ↔ (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆))))
14	9, 13	imbi12d 344	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((⟨𝑠, 𝐹⟩ ∈ dom D → (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑠))) ↔ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)))))
15	1	dmmpossx 8012	. . . . . . . . . 10 ⊢ dom D ⊆ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠))
16	15	sseli 3918	. . . . . . . . 9 ⊢ (⟨𝑠, 𝐹⟩ ∈ dom D → ⟨𝑠, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)))
17		opeliunxp 5691	. . . . . . . . 9 ⊢ (⟨𝑠, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)) ↔ (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑠)))
18	16, 17	sylib 218	. . . . . . . 8 ⊢ (⟨𝑠, 𝐹⟩ ∈ dom D → (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑠)))
19	14, 18	vtoclg 3500	. . . . . . 7 ⊢ (𝑆 ∈ V → (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆))))
20	7, 19	mpcom 38	. . . . . 6 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
21	20	simpld 494	. . . . 5 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ 𝒫 ℂ)
22	21	elpwid 4551	. . . 4 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ⊆ ℂ)
23	20	simprd 495	. . . . . 6 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑_pm 𝑆))
24		cnex 11110	. . . . . . 7 ⊢ ℂ ∈ V
25		elpm2g 8784	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
26	24, 21, 25	sylancr 588	. . . . . 6 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
27	23, 26	mpbid 232	. . . . 5 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
28	27	simpld 494	. . . 4 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹:dom 𝐹⟶ℂ)
29	27	simprd 495	. . . 4 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom 𝐹 ⊆ 𝑆)
30	22, 28, 29	dvbss 25878	. . 3 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) ⊆ dom 𝐹)
31	30, 29	sstrd 3933	. 2 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) ⊆ 𝑆)
32		df-ov 7363	. . . . . 6 ⊢ (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
33		ndmfv 6866	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
34	32, 33	eqtrid 2784	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
35	34	dmeqd 5854	. . . 4 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
36		dm0 5869	. . . 4 ⊢ dom ∅ = ∅
37	35, 36	eqtrdi 2788	. . 3 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
38		0ss 4341	. . 3 ⊢ ∅ ⊆ 𝑆
39	37, 38	eqsstrdi 3967	. 2 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) ⊆ 𝑆)
40	31, 39	pm2.61i 182	1 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ∖ cdif 3887 ⊆ wss 3890 ∅c0 4274 𝒫 cpw 4542 {csn 4568 ⟨cop 4574 ∪ ciun 4934 ↦ cmpt 5167 × cxp 5622 dom cdm 5624 Rel wrel 5629 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ↑_pm cpm 8767 ℂcc 11027 − cmin 11368 / cdiv 11798 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21344 intcnt 22992 lim_ℂ climc 25839 D cdv 25840

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-ntr 22995 df-cnp 23203 df-xms 24295 df-ms 24296 df-limc 25843 df-dv 25844

This theorem is referenced by: dvaddf 25919 dvmulf 25920 dvcmulf 25922 dvcof 25925 dvmptres2 25939 dvmptcmul 25941 dvmptcj 25945 dvcnvlem 25953 dvcnv 25954 dvef 25957 dvcnvrelem1 25994 dvcnvrelem2 25995 dvcnvre 25996 ulmdvlem1 26378 ulmdvlem3 26380 ulmdv 26381 fperdvper 46365 dvmulcncf 46371 dvdivcncf 46373

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