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Theorem rlimcn1b 14728
 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 6664 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵)))
42fmpttd 6649 . . 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 14727 . 2 (𝜑 → (𝐹 ∘ (𝑘𝐴𝐵)) ⇝𝑟 (𝐹𝐶))
93, 8eqbrtrrd 4910 1 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)) ⇝𝑟 (𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   class class class wbr 4886   ↦ cmpt 4965   ∘ ccom 5359  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  ℂcc 10270   < clt 10411   − cmin 10606  ℝ+crp 12137  abscabs 14381   ⇝𝑟 crli 14624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-pm 8143  df-rlim 14628 This theorem is referenced by:  rlimabs  14747  rlimcj  14748  rlimre  14749  rlimim  14750
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