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Theorem rlimres2 15537
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
Hypotheses
Ref Expression
rlimres2.1 (𝜑𝐴𝐵)
rlimres2.2 (𝜑 → (𝑥𝐵𝐶) ⇝𝑟 𝐷)
Assertion
Ref Expression
rlimres2 (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem rlimres2
StepHypRef Expression
1 rlimres2.1 . . 3 (𝜑𝐴𝐵)
21resmptd 6039 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 rlimres2.2 . . 3 (𝜑 → (𝑥𝐵𝐶) ⇝𝑟 𝐷)
4 rlimres 15534 . . 3 ((𝑥𝐵𝐶) ⇝𝑟 𝐷 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
53, 4syl 17 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
62, 5eqbrtrrd 5167 1 (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3939   class class class wbr 5143  cmpt 5226  cres 5674  𝑟 crli 15461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-pm 8846  df-rlim 15465
This theorem is referenced by:  divcnv  15831  dvfsumrlimge0  25983  dvfsumrlim2  25985  dfef2  26921  cxp2lim  26927  chtppilimlem2  27425  chpchtlim  27430  pnt2  27564
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