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Mirrors > Home > ILE Home > Th. List > climrel | GIF version |
Description: The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.) |
Ref | Expression |
---|---|
climrel | ⊢ Rel ⇝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clim 11041 | . 2 ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} | |
2 | 1 | relopabi 4660 | 1 ⊢ Rel ⇝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∈ wcel 1480 ∀wral 2414 ∃wrex 2415 class class class wbr 3924 Rel wrel 4539 ‘cfv 5118 (class class class)co 5767 ℂcc 7611 < clt 7793 − cmin 7926 ℤcz 9047 ℤ≥cuz 9319 ℝ+crp 9434 abscabs 10762 ⇝ cli 11040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-rel 4541 df-clim 11041 |
This theorem is referenced by: clim 11043 climcl 11044 climi 11049 fclim 11056 climrecl 11086 iserex 11101 climrecvg1n 11110 climcvg1nlem 11111 fsum3cvg3 11158 trirecip 11263 ntrivcvgap0 11311 |
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