![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 15522 | . 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 5833 | 1 ⊢ Rel ⇝𝑟 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 dom cdm 5689 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 ↑pm cpm 8866 ℂcc 11151 ℝcr 11152 < clt 11293 ≤ cle 11294 − cmin 11490 ℝ+crp 13032 abscabs 15270 ⇝𝑟 crli 15518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-opab 5211 df-xp 5695 df-rel 5696 df-rlim 15522 |
This theorem is referenced by: rlim 15528 rlimpm 15533 rlimdm 15584 caucvgrlem2 15708 caucvgr 15709 rlimdmafv 47127 rlimdmafv2 47208 |
Copyright terms: Public domain | W3C validator |