Theorem limcresi 15460

Description: Any limit of 𝐹 is also a limit of the restriction of 𝐹. (Contributed by Mario Carneiro, 28-Dec-2016.)

Assertion

Ref	Expression
limcresi	⊢ (𝐹 limℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) limℂ 𝐵)

Proof of Theorem limcresi

Dummy variables 𝑑 𝑒 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limcrcl 15452	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
2	1	simp1d 1036	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐹:dom 𝐹⟶ℂ)
3	1	simp2d 1037	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → dom 𝐹 ⊆ ℂ)
4	1	simp3d 1038	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
5	2, 3, 4	ellimc3ap 15455	. . . . 5 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒))))
6	5	ibi 176	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒)))
7		inss1 3429	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ 𝐶) ⊆ dom 𝐹
8		ssralv 3292	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ 𝐶) ⊆ dom 𝐹 → (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒)))
9	7, 8	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒))
10		elinel2 3396	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (dom 𝐹 ∩ 𝐶) → 𝑢 ∈ 𝐶)
11		fvres 5672	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑢) = (𝐹‘𝑢))
12	10, 11	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (dom 𝐹 ∩ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑢) = (𝐹‘𝑢))
13	12	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → ((𝐹 ↾ 𝐶)‘𝑢) = (𝐹‘𝑢))
14	13	fvoveq1d 6050	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) = (abs‘((𝐹‘𝑢) − 𝑥)))
15	14	breq1d 4103	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → ((abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒 ↔ (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒))
16	15	imbi2d 230	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → (((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒) ↔ ((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒)))
17	16	biimprd 158	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → (((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
18	17	ralimdva 2600	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
19	9, 18	syl5 32	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
20	19	reximdv 2634	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
21	20	ralimdv 2601	. . . . 5 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒) → ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
22	21	anim2d 337	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → ((𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘((𝐹‘𝑢) − 𝑥)) < 𝑒)) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒))))
23	6, 22	mpd 13	. . 3 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒)))
24		fresin 5523	. . . . 5 ⊢ (𝐹:dom 𝐹⟶ℂ → (𝐹 ↾ 𝐶):(dom 𝐹 ∩ 𝐶)⟶ℂ)
25	2, 24	syl 14	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹 ↾ 𝐶):(dom 𝐹 ∩ 𝐶)⟶ℂ)
26	7, 3	sstrid 3239	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (dom 𝐹 ∩ 𝐶) ⊆ ℂ)
27	25, 26, 4	ellimc3ap 15455	. . 3 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝑥 ∈ ((𝐹 ↾ 𝐶) limℂ 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ (dom 𝐹 ∩ 𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢 − 𝐵)) < 𝑑) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) < 𝑒))))
28	23, 27	mpbird 167	. 2 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝑥 ∈ ((𝐹 ↾ 𝐶) limℂ 𝐵))
29	28	ssriv 3232	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) limℂ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512   ∩ cin 3200   ⊆ wss 3201   class class class wbr 4093  dom cdm 4731   ↾ cres 4733  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  ℂcc 8073   < clt 8256   − cmin 8392   # cap 8803  ℝ⁺crp 9932  abscabs 11620   lim_ℂ climc 15448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pm 6863  df-limced 15450

This theorem is referenced by:  dvidlemap  15485  dvidrelem  15486  dvidsslem  15487  dvcnp2cntop  15493  dvcoapbr  15501