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Theorem rlimres2 15511
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 6034 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 rlimres2.2 . . 3 (𝜑 → (𝑥𝐵𝐶) ⇝𝑟 𝐷)
4 rlimres 15508 . . 3 ((𝑥𝐵𝐶) ⇝𝑟 𝐷 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
53, 4syl 17 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
62, 5eqbrtrrd 5165 1 (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3943   class class class wbr 5141  cmpt 5224  cres 5671  𝑟 crli 15435
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-pm 8825  df-rlim 15439
This theorem is referenced by:  divcnv  15805  dvfsumrlimge0  25920  dvfsumrlim2  25922  dfef2  26858  cxp2lim  26864  chtppilimlem2  27362  chpchtlim  27367  pnt2  27501
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