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Theorem rlimres 14218
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimres (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)

Proof of Theorem rlimres
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3816 . . . . . . . 8 (dom 𝐹𝐵) ⊆ dom 𝐹
2 ssralv 3650 . . . . . . . 8 ((dom 𝐹𝐵) ⊆ dom 𝐹 → (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)))
31, 2ax-mp 5 . . . . . . 7 (∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
43reximi 3010 . . . . . 6 (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
54ralimi 2952 . . . . 5 (∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥) → ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))
65anim2i 592 . . . 4 ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)))
76a1i 11 . . 3 (𝐹𝑟 𝐴 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥)) → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
8 rlimf 14161 . . . 4 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
9 rlimss 14162 . . . 4 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
10 eqidd 2627 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ dom 𝐹) → (𝐹𝑧) = (𝐹𝑧))
118, 9, 10rlim 14155 . . 3 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
12 fssres 6029 . . . . . 6 ((𝐹:dom 𝐹⟶ℂ ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
138, 1, 12sylancl 693 . . . . 5 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ)
14 resres 5372 . . . . . . 7 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
15 ffn 6004 . . . . . . . . 9 (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹)
16 fnresdm 5960 . . . . . . . . 9 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
178, 15, 163syl 18 . . . . . . . 8 (𝐹𝑟 𝐴 → (𝐹 ↾ dom 𝐹) = 𝐹)
1817reseq1d 5359 . . . . . . 7 (𝐹𝑟 𝐴 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1914, 18syl5eqr 2674 . . . . . 6 (𝐹𝑟 𝐴 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2019feq1d 5989 . . . . 5 (𝐹𝑟 𝐴 → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶ℂ ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ))
2113, 20mpbid 222 . . . 4 (𝐹𝑟 𝐴 → (𝐹𝐵):(dom 𝐹𝐵)⟶ℂ)
221, 9syl5ss 3599 . . . 4 (𝐹𝑟 𝐴 → (dom 𝐹𝐵) ⊆ ℝ)
23 inss2 3817 . . . . . . 7 (dom 𝐹𝐵) ⊆ 𝐵
2423sseli 3584 . . . . . 6 (𝑧 ∈ (dom 𝐹𝐵) → 𝑧𝐵)
25 fvres 6165 . . . . . 6 (𝑧𝐵 → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2624, 25syl 17 . . . . 5 (𝑧 ∈ (dom 𝐹𝐵) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2726adantl 482 . . . 4 ((𝐹𝑟 𝐴𝑧 ∈ (dom 𝐹𝐵)) → ((𝐹𝐵)‘𝑧) = (𝐹𝑧))
2821, 22, 27rlim 14155 . . 3 (𝐹𝑟 𝐴 → ((𝐹𝐵) ⇝𝑟 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℝ ∀𝑧 ∈ (dom 𝐹𝐵)(𝑦𝑧 → (abs‘((𝐹𝑧) − 𝐴)) < 𝑥))))
297, 11, 283imtr4d 283 . 2 (𝐹𝑟 𝐴 → (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴))
3029pm2.43i 52 1 (𝐹𝑟 𝐴 → (𝐹𝐵) ⇝𝑟 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cin 3559  wss 3560   class class class wbr 4618  dom cdm 5079  cres 5081   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  cc 9879  cr 9880   < clt 10019  cle 10020  cmin 10211  +crp 11776  abscabs 13903  𝑟 crli 14145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-pm 7806  df-rlim 14149
This theorem is referenced by:  rlimres2  14221  pnt  25198
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