Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9800 | . 2 ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} | |
2 | 1 | relopabi 4463 | 1 ⊢ Rel ⇝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∈ wcel 1393 ∀wral 2306 ∃wrex 2307 class class class wbr 3764 Rel wrel 4350 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 < clt 7060 − cmin 7182 ℤcz 8245 ℤ≥cuz 8473 ℝ+crp 8583 abscabs 9595 ⇝ cli 9799 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 df-rel 4352 df-clim 9800 |
This theorem is referenced by: clim 9802 climcl 9803 climi 9808 fclim 9815 climrecl 9844 iiserex 9859 climrecvg1n 9867 climcvg1nlem 9868 |
Copyright terms: Public domain | W3C validator |