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Mirrors > Home > MPE Home > Th. List > climrel | Structured version Visualization version 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 14420 | . 2 ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} | |
2 | 1 | relopabi 5382 | 1 ⊢ Rel ⇝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 class class class wbr 4786 Rel wrel 5254 ‘cfv 6029 (class class class)co 6791 ℂcc 10134 < clt 10274 − cmin 10466 ℤcz 11577 ℤ≥cuz 11886 ℝ+crp 12028 abscabs 14175 ⇝ cli 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-opab 4847 df-xp 5255 df-rel 5256 df-clim 14420 |
This theorem is referenced by: clim 14426 climcl 14431 climi 14442 climrlim2 14479 fclim 14485 climrecl 14515 climge0 14516 iserex 14588 caurcvg2 14609 caucvg 14610 iseralt 14616 fsumcvg3 14661 cvgcmpce 14750 climfsum 14752 climcnds 14783 trirecip 14795 ntrivcvgn0 14830 ovoliunlem1 23483 mbflimlem 23647 abelthlem5 24402 emcllem6 24941 lgamgulmlem4 24972 binomcxplemnn0 39067 binomcxplemnotnn0 39074 climf 40365 sumnnodd 40373 climf2 40409 climd 40415 clim2d 40416 climfv 40434 climuzlem 40486 climlimsup 40503 climlimsupcex 40512 climliminflimsupd 40544 climliminf 40549 liminflimsupclim 40550 ioodvbdlimc1lem2 40658 ioodvbdlimc2lem 40660 stirlinglem12 40812 fouriersw 40958 |
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