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Theorem rlimcn1b 14254
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.1 . . 3 ((𝜑𝑘𝐴) → 𝐵𝑋)
2 eqidd 2622 . . 3 (𝜑 → (𝑘𝐴𝐵) = (𝑘𝐴𝐵))
3 rlimcn1b.4 . . . 4 (𝜑𝐹:𝑋⟶ℂ)
43feqmptd 6206 . . 3 (𝜑𝐹 = (𝑧𝑋 ↦ (𝐹𝑧)))
5 fveq2 6148 . . 3 (𝑧 = 𝐵 → (𝐹𝑧) = (𝐹𝐵))
61, 2, 4, 5fmptco 6351 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵)))
7 eqid 2621 . . . 4 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
81, 7fmptd 6340 . . 3 (𝜑 → (𝑘𝐴𝐵):𝐴𝑋)
9 rlimcn1b.2 . . 3 (𝜑𝐶𝑋)
10 rlimcn1b.3 . . 3 (𝜑 → (𝑘𝐴𝐵) ⇝𝑟 𝐶)
11 rlimcn1b.5 . . 3 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+𝑧𝑋 ((abs‘(𝑧𝐶)) < 𝑦 → (abs‘((𝐹𝑧) − (𝐹𝐶))) < 𝑥))
128, 9, 10, 3, 11rlimcn1 14253 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) ⇝𝑟 (𝐹𝐶))
136, 12eqbrtrrd 4637 1 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  cmpt 4673  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cc 9878   < clt 10018  cmin 10210  +crp 11776  abscabs 13908  𝑟 crli 14150
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-pm 7805  df-rlim 14154
This theorem is referenced by:  rlimabs  14273  rlimcj  14274  rlimre  14275  rlimim  14276
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