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Theorem rlimcn1b 15622
Description: Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)
Hypotheses
Ref Expression
rlimcn1b.1 ((𝜑𝑘𝐴) → 𝐵𝑋)
rlimcn1b.2 (𝜑𝐶𝑋)
rlimcn1b.3 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
rlimcn1b.4 (𝜑𝐹:𝑋⟶ℂ)
rlimcn1b.5 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
Assertion
Ref Expression
rlimcn1b (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
Distinct variable groups:   𝑥,𝑘,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑘,𝐹,𝑥,𝑦,𝑧   𝑘,𝑋,𝑧   𝜑,𝑘,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑘)   𝐶(𝑘)   𝑋(𝑥,𝑦)

Proof of Theorem rlimcn1b
StepHypRef Expression
1 rlimcn1b.4 . . 3 (𝜑𝐹:𝑋⟶ℂ)
2 rlimcn1b.1 . . 3 ((𝜑𝑘𝐴) → 𝐵𝑋)
31, 2cofmpt 7152 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵)))
42fmpttd 7135 . . 3 (𝜑 → (𝑘𝐴𝐵):𝐴𝑋)
5 rlimcn1b.2 . . 3 (𝜑𝐶𝑋)
6 rlimcn1b.3 . . 3 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
7 rlimcn1b.5 . . 3 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
84, 5, 6, 1, 7rlimcn1 15621 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) ⇝𝑟 (𝐹𝐶))
93, 8eqbrtrrd 5172 1 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wrex 3068   class class class wbr 5148  cmpt 5231  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  cc 11151   < clt 11293  cmin 11490  +crp 13032  abscabs 15270  𝑟 crli 15518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8868  df-rlim 15522
This theorem is referenced by:  rlimabs  15642  rlimcj  15643  rlimre  15644  rlimim  15645
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