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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 11080 | . 2 ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} | |
2 | 1 | relopabi 4673 | 1 ⊢ Rel ⇝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 class class class wbr 3937 Rel wrel 4552 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 < clt 7824 − cmin 7957 ℤcz 9078 ℤ≥cuz 9350 ℝ+crp 9470 abscabs 10801 ⇝ cli 11079 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 df-rel 4554 df-clim 11080 |
This theorem is referenced by: clim 11082 climcl 11083 climi 11088 fclim 11095 climrecl 11125 iserex 11140 climrecvg1n 11149 climcvg1nlem 11150 fsum3cvg3 11197 trirecip 11302 ntrivcvgap0 11350 |
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