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Theorem rlimres2 14910
 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 5901 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
3 rlimres2.2 . . 3 (𝜑 → (𝑥𝐵𝐶) ⇝𝑟 𝐷)
4 rlimres 14907 . . 3 ((𝑥𝐵𝐶) ⇝𝑟 𝐷 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
53, 4syl 17 . 2 (𝜑 → ((𝑥𝐵𝐶) ↾ 𝐴) ⇝𝑟 𝐷)
62, 5eqbrtrrd 5081 1 (𝜑 → (𝑥𝐴𝐶) ⇝𝑟 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3934   class class class wbr 5057   ↦ cmpt 5137   ↾ cres 5550   ⇝𝑟 crli 14834 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-pm 8401  df-rlim 14838 This theorem is referenced by:  divcnv  15200  dvfsumrlimge0  24619  dvfsumrlim2  24621  dfef2  25540  cxp2lim  25546  chtppilimlem2  26042  chpchtlim  26047  pnt2  26181
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