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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 15459 | . 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 5816 | 1 ⊢ Rel ⇝𝑟 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 class class class wbr 5142 dom cdm 5672 Rel wrel 5677 ‘cfv 6542 (class class class)co 7414 ↑pm cpm 8839 ℂcc 11130 ℝcr 11131 < clt 11272 ≤ cle 11273 − cmin 11468 ℝ+crp 13000 abscabs 15207 ⇝𝑟 crli 15455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-ss 3962 df-opab 5205 df-xp 5678 df-rel 5679 df-rlim 15459 |
This theorem is referenced by: rlim 15465 rlimpm 15470 rlimdm 15521 caucvgrlem2 15647 caucvgr 15648 rlimdmafv 46551 rlimdmafv2 46632 |
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