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Mirrors > Home > MPE Home > Th. List > rlimrel | Structured version Visualization version GIF version |
Description: The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.) |
Ref | Expression |
---|---|
rlimrel | ⊢ Rel ⇝𝑟 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rlim 14596 | . 2 ⊢ ⇝𝑟 = {〈𝑓, 𝑥〉 ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))} | |
2 | 1 | relopabi 5477 | 1 ⊢ Rel ⇝𝑟 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 ∀wral 3116 ∃wrex 3117 class class class wbr 4872 dom cdm 5341 Rel wrel 5346 ‘cfv 6122 (class class class)co 6904 ↑pm cpm 8122 ℂcc 10249 ℝcr 10250 < clt 10390 ≤ cle 10391 − cmin 10584 ℝ+crp 12111 abscabs 14350 ⇝𝑟 crli 14592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-rab 3125 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-opab 4935 df-xp 5347 df-rel 5348 df-rlim 14596 |
This theorem is referenced by: rlim 14602 rlimpm 14607 rlimdm 14658 caucvgrlem2 14781 caucvgr 14782 rlimdmafv 42078 rlimdmafv2 42159 |
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