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| 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 15526 | . 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 5829 | 1 ⊢ Rel ⇝𝑟 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 class class class wbr 5142 dom cdm 5684 Rel wrel 5689 ‘cfv 6560 (class class class)co 7432 ↑pm cpm 8868 ℂcc 11154 ℝcr 11155 < clt 11296 ≤ cle 11297 − cmin 11493 ℝ+crp 13035 abscabs 15274 ⇝𝑟 crli 15522 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-opab 5205 df-xp 5690 df-rel 5691 df-rlim 15526 | 
| This theorem is referenced by: rlim 15532 rlimpm 15537 rlimdm 15588 caucvgrlem2 15712 caucvgr 15713 rlimdmafv 47194 rlimdmafv2 47275 | 
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