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Theorem rlimcn1b 15593
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 7148 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵)))
42fmpttd 7131 . . 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 15592 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) ⇝𝑟 (𝐹𝐶))
93, 8eqbrtrrd 5179 1 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  wral 3051  wrex 3060   class class class wbr 5155  cmpt 5238  ccom 5688  wf 6552  cfv 6556  (class class class)co 7426  cc 11158   < clt 11300  cmin 11496  +crp 13030  abscabs 15241  𝑟 crli 15489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fv 6564  df-ov 7429  df-oprab 7430  df-mpo 7431  df-pm 8860  df-rlim 15493
This theorem is referenced by:  rlimabs  15613  rlimcj  15614  rlimre  15615  rlimim  15616
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