Theorem limcresi 15690

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 15682	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
2	1	simp1d 1040	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐹:dom 𝐹⟶ℂ)
3	1	simp2d 1041	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → dom 𝐹 ⊆ ℂ)
4	1	simp3d 1042	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
5	2, 3, 4	ellimc3ap 15685	. . . . 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 3451	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ 𝐶) ⊆ dom 𝐹
8		ssralv 3312	. . . . . . . . 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 3416	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (dom 𝐹 ∩ 𝐶) → 𝑢 ∈ 𝐶)
11		fvres 5714	. . . . . . . . . . . . . . 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 6097	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) = (abs‘((𝐹‘𝑢) − 𝑥)))
15	14	breq1d 4135	. . . . . . . . . . 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 2617	. . . . . . . 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 2651	. . . . . 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 2618	. . . . 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 5563	. . . . 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 3259	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (dom 𝐹 ∩ 𝐶) ⊆ ℂ)
27	25, 26, 4	ellimc3ap 15685	. . 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 3252	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) limℂ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ∩ cin 3219   ⊆ wss 3220   class class class wbr 4125  dom cdm 4769   ↾ cres 4771  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  ℂcc 8167   < clt 8350   − cmin 8487   # cap 8899  ℝ⁺crp 10033  abscabs 11741   lim_ℂ climc 15678

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pm 6915  df-limced 15680

This theorem is referenced by:  dvidlemap  15715  dvidrelem  15716  dvidsslem  15717  dvcnp2cntop  15723  dvcoapbr  15731