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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 15063 | . 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 | relopabiv 5699 | 1 ⊢ Rel ⇝𝑟 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3062 ∃wrex 3063 class class class wbr 5062 dom cdm 5560 Rel wrel 5565 ‘cfv 6389 (class class class)co 7222 ↑pm cpm 8518 ℂcc 10740 ℝcr 10741 < clt 10880 ≤ cle 10881 − cmin 11075 ℝ+crp 12599 abscabs 14810 ⇝𝑟 crli 15059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-v 3417 df-in 3882 df-ss 3892 df-opab 5125 df-xp 5566 df-rel 5567 df-rlim 15063 |
This theorem is referenced by: rlim 15069 rlimpm 15074 rlimdm 15125 caucvgrlem2 15251 caucvgr 15252 rlimdmafv 44356 rlimdmafv2 44437 |
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