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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 14560 | . 2 ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} | |
2 | 1 | relopabi 5449 | 1 ⊢ Rel ⇝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 ∈ wcel 2157 ∀wral 3089 ∃wrex 3090 class class class wbr 4843 Rel wrel 5317 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 < clt 10363 − cmin 10556 ℤcz 11666 ℤ≥cuz 11930 ℝ+crp 12074 abscabs 14315 ⇝ cli 14556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-opab 4906 df-xp 5318 df-rel 5319 df-clim 14560 |
This theorem is referenced by: clim 14566 climcl 14571 climi 14582 climrlim2 14619 fclim 14625 climrecl 14655 climge0 14656 iserex 14728 caurcvg2 14749 caucvg 14750 iseralt 14756 fsumcvg3 14801 cvgcmpce 14888 climfsum 14890 climcnds 14921 trirecip 14933 ntrivcvgn0 14967 ovoliunlem1 23610 mbflimlem 23775 abelthlem5 24530 emcllem6 25079 lgamgulmlem4 25110 binomcxplemnn0 39330 binomcxplemnotnn0 39337 climf 40598 sumnnodd 40606 climf2 40642 climd 40648 clim2d 40649 climfv 40667 climuzlem 40719 climlimsup 40736 climlimsupcex 40745 climliminflimsupd 40777 climliminf 40782 liminflimsupclim 40783 ioodvbdlimc1lem2 40891 ioodvbdlimc2lem 40893 stirlinglem12 41045 fouriersw 41191 |
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