Theorem limcresi 15138

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 15130	. . . . . . 7 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
2	1	simp1d 1012	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐹:dom 𝐹⟶ℂ)
3	1	simp2d 1013	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → dom 𝐹 ⊆ ℂ)
4	1	simp3d 1014	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → 𝐵 ∈ ℂ)
5	2, 3, 4	ellimc3ap 15133	. . . . 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 3393	. . . . . . . . 9 ⊢ (dom 𝐹 ∩ 𝐶) ⊆ dom 𝐹
8		ssralv 3257	. . . . . . . . 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 3360	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (dom 𝐹 ∩ 𝐶) → 𝑢 ∈ 𝐶)
11		fvres 5600	. . . . . . . . . . . . . . 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 5966	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝐹 limℂ 𝐵) ∧ 𝑢 ∈ (dom 𝐹 ∩ 𝐶)) → (abs‘(((𝐹 ↾ 𝐶)‘𝑢) − 𝑥)) = (abs‘((𝐹‘𝑢) − 𝑥)))
15	14	breq1d 4054	. . . . . . . . . . 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 2573	. . . . . . . 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 2607	. . . . . 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 2574	. . . . 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 5454	. . . . 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 3204	. . . 4 ⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (dom 𝐹 ∩ 𝐶) ⊆ ℂ)
27	25, 26, 4	ellimc3ap 15133	. . 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 3197	1 ⊢ (𝐹 limℂ 𝐵) ⊆ ((𝐹 ↾ 𝐶) limℂ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485   ∩ cin 3165   ⊆ wss 3166   class class class wbr 4044  dom cdm 4675   ↾ cres 4677  ⟶wf 5267  ‘cfv 5271  (class class class)co 5944  ℂcc 7923   < clt 8107   − cmin 8243   # cap 8654  ℝ⁺crp 9775  abscabs 11308   lim_ℂ climc 15126

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pm 6738  df-limced 15128

This theorem is referenced by:  dvidlemap  15163  dvidrelem  15164  dvidsslem  15165  dvcnp2cntop  15171  dvcoapbr  15179