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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 6058 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | 
| 3 | rlimres2.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝𝑟 𝐷) | |
| 4 | rlimres 15594 | . . 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 5167 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ⇝𝑟 crli 15521 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 df-rlim 15525 | 
| This theorem is referenced by: divcnv 15889 dvfsumrlimge0 26071 dvfsumrlim2 26073 dfef2 27014 cxp2lim 27020 chtppilimlem2 27518 chpchtlim 27523 pnt2 27657 | 
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