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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 14068 | . 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 5167 | 1 ⊢ Rel ⇝𝑟 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 ↑pm cpm 7745 ℂcc 9813 ℝcr 9814 < clt 9953 ≤ cle 9954 − cmin 10145 ℝ+crp 11708 abscabs 13822 ⇝𝑟 crli 14064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-rlim 14068 |
This theorem is referenced by: rlim 14074 rlimpm 14079 rlimdm 14130 caucvgrlem2 14253 caucvgr 14254 rlimdmafv 39906 |
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