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Theorem rlimcn1b 15625
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 15624 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) ⇝𝑟 (𝐹𝐶))
93, 8eqbrtrrd 5167 1 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  cmpt 5225  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cc 11153   < clt 11295  cmin 11492  +crp 13034  abscabs 15273  𝑟 crli 15521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-pm 8869  df-rlim 15525
This theorem is referenced by:  rlimabs  15645  rlimcj  15646  rlimre  15647  rlimim  15648
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