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Theorem limcresi 12830
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 lim 𝐵) ⊆ ((𝐹𝐶) lim 𝐵)

Proof of Theorem limcresi
Dummy variables 𝑑 𝑒 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limcrcl 12822 . . . . . . 7 (𝑥 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
21simp1d 993 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → 𝐹:dom 𝐹⟶ℂ)
31simp2d 994 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → dom 𝐹 ⊆ ℂ)
41simp3d 995 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → 𝐵 ∈ ℂ)
52, 3, 4ellimc3ap 12825 . . . . 5 (𝑥 ∈ (𝐹 lim 𝐵) → (𝑥 ∈ (𝐹 lim 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒))))
65ibi 175 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒)))
7 inss1 3296 . . . . . . . . 9 (dom 𝐹𝐶) ⊆ dom 𝐹
8 ssralv 3161 . . . . . . . . 9 ((dom 𝐹𝐶) ⊆ dom 𝐹 → (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒)))
97, 8ax-mp 5 . . . . . . . 8 (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒))
10 elinel2 3263 . . . . . . . . . . . . . . 15 (𝑢 ∈ (dom 𝐹𝐶) → 𝑢𝐶)
11 fvres 5448 . . . . . . . . . . . . . . 15 (𝑢𝐶 → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1210, 11syl 14 . . . . . . . . . . . . . 14 (𝑢 ∈ (dom 𝐹𝐶) → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1312adantl 275 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1413fvoveq1d 5799 . . . . . . . . . . . 12 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) = (abs‘((𝐹𝑢) − 𝑥)))
1514breq1d 3942 . . . . . . . . . . 11 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → ((abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒 ↔ (abs‘((𝐹𝑢) − 𝑥)) < 𝑒))
1615imbi2d 229 . . . . . . . . . 10 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → (((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒) ↔ ((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒)))
1716biimprd 157 . . . . . . . . 9 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → (((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
1817ralimdva 2499 . . . . . . . 8 (𝑥 ∈ (𝐹 lim 𝐵) → (∀𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
199, 18syl5 32 . . . . . . 7 (𝑥 ∈ (𝐹 lim 𝐵) → (∀𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∀𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
2019reximdv 2533 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → (∃𝑑 ∈ ℝ+𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∃𝑑 ∈ ℝ+𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
2120ralimdv 2500 . . . . 5 (𝑥 ∈ (𝐹 lim 𝐵) → (∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒) → ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
2221anim2d 335 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → ((𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ dom 𝐹((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘((𝐹𝑢) − 𝑥)) < 𝑒)) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒))))
236, 22mpd 13 . . 3 (𝑥 ∈ (𝐹 lim 𝐵) → (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒)))
24 fresin 5304 . . . . 5 (𝐹:dom 𝐹⟶ℂ → (𝐹𝐶):(dom 𝐹𝐶)⟶ℂ)
252, 24syl 14 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → (𝐹𝐶):(dom 𝐹𝐶)⟶ℂ)
267, 3sstrid 3108 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → (dom 𝐹𝐶) ⊆ ℂ)
2725, 26, 4ellimc3ap 12825 . . 3 (𝑥 ∈ (𝐹 lim 𝐵) → (𝑥 ∈ ((𝐹𝐶) lim 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑢 ∈ (dom 𝐹𝐶)((𝑢 # 𝐵 ∧ (abs‘(𝑢𝐵)) < 𝑑) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒))))
2823, 27mpbird 166 . 2 (𝑥 ∈ (𝐹 lim 𝐵) → 𝑥 ∈ ((𝐹𝐶) lim 𝐵))
2928ssriv 3101 1 (𝐹 lim 𝐵) ⊆ ((𝐹𝐶) lim 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wral 2416  wrex 2417  cin 3070  wss 3071   class class class wbr 3932  dom cdm 4542  cres 4544  wf 5122  cfv 5126  (class class class)co 5777  cc 7637   < clt 7819  cmin 7952   # cap 8362  +crp 9463  abscabs 10793   lim climc 12818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-cnex 7730
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-pm 6548  df-limced 12820
This theorem is referenced by:  dvidlemap  12855  dvcnp2cntop  12858  dvcoapbr  12866
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