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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  ℂcc 7637   < clt 7819   − cmin 7952   # cap 8362  ℝ+crp 9463  abscabs 10793   limℂ 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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