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Mirrors > Home > MPE Home > Th. List > rlimres2 | Structured version Visualization version GIF version |
Description: The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.) |
Ref | Expression |
---|---|
rlimres2.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
rlimres2.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝𝑟 𝐷) |
Ref | Expression |
---|---|
rlimres2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimres2.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | 1 | resmptd 5910 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
3 | rlimres2.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝𝑟 𝐷) | |
4 | rlimres 14917 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↦ 𝐶) ⇝𝑟 𝐷 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⇝𝑟 𝐷) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) ⇝𝑟 𝐷) |
6 | 2, 5 | eqbrtrrd 5092 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ↾ cres 5559 ⇝𝑟 crli 14844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-pm 8411 df-rlim 14848 |
This theorem is referenced by: divcnv 15210 dvfsumrlimge0 24629 dvfsumrlim2 24631 dfef2 25550 cxp2lim 25556 chtppilimlem2 26052 chpchtlim 26057 pnt2 26191 |
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