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Theorem limcresi 13795
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 13787 . . . . . . 7 (𝑥 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
21simp1d 1009 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → 𝐹:dom 𝐹⟶ℂ)
31simp2d 1010 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → dom 𝐹 ⊆ ℂ)
41simp3d 1011 . . . . . 6 (𝑥 ∈ (𝐹 lim 𝐵) → 𝐵 ∈ ℂ)
52, 3, 4ellimc3ap 13790 . . . . 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 176 . . . 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 3355 . . . . . . . . 9 (dom 𝐹𝐶) ⊆ dom 𝐹
8 ssralv 3219 . . . . . . . . 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 3322 . . . . . . . . . . . . . . 15 (𝑢 ∈ (dom 𝐹𝐶) → 𝑢𝐶)
11 fvres 5535 . . . . . . . . . . . . . . 15 (𝑢𝐶 → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1210, 11syl 14 . . . . . . . . . . . . . 14 (𝑢 ∈ (dom 𝐹𝐶) → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1312adantl 277 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → ((𝐹𝐶)‘𝑢) = (𝐹𝑢))
1413fvoveq1d 5891 . . . . . . . . . . . 12 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → (abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) = (abs‘((𝐹𝑢) − 𝑥)))
1514breq1d 4010 . . . . . . . . . . 11 ((𝑥 ∈ (𝐹 lim 𝐵) ∧ 𝑢 ∈ (dom 𝐹𝐶)) → ((abs‘(((𝐹𝐶)‘𝑢) − 𝑥)) < 𝑒 ↔ (abs‘((𝐹𝑢) − 𝑥)) < 𝑒))
1615imbi2d 230 . . . . . . . . . 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 158 . . . . . . . . 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 2544 . . . . . . . 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 2578 . . . . . 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 2545 . . . . 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 337 . . . 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 5390 . . . . 5 (𝐹:dom 𝐹⟶ℂ → (𝐹𝐶):(dom 𝐹𝐶)⟶ℂ)
252, 24syl 14 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → (𝐹𝐶):(dom 𝐹𝐶)⟶ℂ)
267, 3sstrid 3166 . . . 4 (𝑥 ∈ (𝐹 lim 𝐵) → (dom 𝐹𝐶) ⊆ ℂ)
2725, 26, 4ellimc3ap 13790 . . 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 167 . 2 (𝑥 ∈ (𝐹 lim 𝐵) → 𝑥 ∈ ((𝐹𝐶) lim 𝐵))
2928ssriv 3159 1 (𝐹 lim 𝐵) ⊆ ((𝐹𝐶) lim 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wrex 2456  cin 3128  wss 3129   class class class wbr 4000  dom cdm 4623  cres 4625  wf 5208  cfv 5212  (class class class)co 5869  cc 7797   < clt 7979  cmin 8115   # cap 8525  +crp 9637  abscabs 10987   lim climc 13783
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7890
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pm 6645  df-limced 13785
This theorem is referenced by:  dvidlemap  13820  dvcnp2cntop  13823  dvcoapbr  13831
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