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Theorem dvbsss 25850

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 25815	. . . . . . . . . . 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 7489	. . . . . . . . . 10 ⊢ Rel dom D
3		df-rel 5628	. . . . . . . . . 10 ⊢ (Rel dom D ↔ dom D ⊆ (V × V))
4	2, 3	mpbi 230	. . . . . . . . 9 ⊢ dom D ⊆ (V × V)
5	4	sseli 3926	. . . . . . . 8 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → ⟨𝑆, 𝐹⟩ ∈ (V × V))
6		opelxp1 5663	. . . . . . . 8 ⊢ (⟨𝑆, 𝐹⟩ ∈ (V × V) → 𝑆 ∈ V)
7	5, 6	syl 17	. . . . . . 7 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ∈ V)
8		opeq1 4826	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ⟨𝑠, 𝐹⟩ = ⟨𝑆, 𝐹⟩)
9	8	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (⟨𝑠, 𝐹⟩ ∈ dom D ↔ ⟨𝑆, 𝐹⟩ ∈ dom D ))
10		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 ℂ ↔ 𝑆 ∈ 𝒫 ℂ))
11		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (ℂ ↑_pm 𝑠) = (ℂ ↑_pm 𝑆))
12	11	eleq2d 2819	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (ℂ ↑_pm 𝑠) ↔ 𝐹 ∈ (ℂ ↑_pm 𝑆)))
13	10, 12	anbi12d 632	. . . . . . . . 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 8007	. . . . . . . . . 10 ⊢ dom D ⊆ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠))
16	15	sseli 3926	. . . . . . . . 9 ⊢ (⟨𝑠, 𝐹⟩ ∈ dom D → ⟨𝑠, 𝐹⟩ ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑_pm 𝑠)))
17		opeliunxp 5688	. . . . . . . . 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 3508	. . . . . . 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 4560	. . . 4 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝑆 ⊆ ℂ)
23	20	simprd 495	. . . . . 6 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → 𝐹 ∈ (ℂ ↑_pm 𝑆))
24		cnex 11098	. . . . . . 7 ⊢ ℂ ∈ V
25		elpm2g 8777	. . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑_pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)))
26	24, 21, 25	sylancr 587	. . . . . 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 25849	. . 3 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) ⊆ dom 𝐹)
31	30, 29	sstrd 3941	. 2 ⊢ (⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) ⊆ 𝑆)
32		df-ov 7358	. . . . . 6 ⊢ (𝑆 D 𝐹) = ( D ‘⟨𝑆, 𝐹⟩)
33		ndmfv 6863	. . . . . 6 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → ( D ‘⟨𝑆, 𝐹⟩) = ∅)
34	32, 33	eqtrid 2780	. . . . 5 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → (𝑆 D 𝐹) = ∅)
35	34	dmeqd 5851	. . . 4 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = dom ∅)
36		dm0 5866	. . . 4 ⊢ dom ∅ = ∅
37	35, 36	eqtrdi 2784	. . 3 ⊢ (¬ ⟨𝑆, 𝐹⟩ ∈ dom D → dom (𝑆 D 𝐹) = ∅)
38		0ss 4349	. . 3 ⊢ ∅ ⊆ 𝑆
39	37, 38	eqsstrdi 3975	. 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 1541 ∈ wcel 2113 Vcvv 3437 ∖ cdif 3895 ⊆ wss 3898 ∅c0 4282 𝒫 cpw 4551 {csn 4577 ⟨cop 4583 ∪ ciun 4943 ↦ cmpt 5176 × cxp 5619 dom cdm 5621 Rel wrel 5626 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_pm cpm 8760 ℂcc 11015 − cmin 11355 / cdiv 11785 ↾_t crest 17331 TopOpenctopn 17332 ℂ_fldccnfld 21300 intcnt 22952 lim_ℂ climc 25810 D cdv 25811

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9306 df-sup 9337 df-inf 9338 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-fz 13415 df-seq 13916 df-exp 13976 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-struct 17065 df-slot 17100 df-ndx 17112 df-base 17128 df-plusg 17181 df-mulr 17182 df-starv 17183 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-rest 17333 df-topn 17334 df-topgen 17354 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-ntr 22955 df-cnp 23163 df-xms 24255 df-ms 24256 df-limc 25814 df-dv 25815

This theorem is referenced by: dvaddf 25892 dvmulf 25893 dvcmulf 25895 dvcof 25899 dvmptres2 25913 dvmptcmul 25915 dvmptcj 25919 dvcnvlem 25927 dvcnv 25928 dvef 25931 dvcnvrelem1 25969 dvcnvrelem2 25970 dvcnvre 25971 ulmdvlem1 26356 ulmdvlem3 26358 ulmdv 26359 fperdvper 46079 dvmulcncf 46085 dvdivcncf 46087

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