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Mirrors > Home > MPE Home > Th. List > limcrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the limit operator. (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
limcrcl | β’ (πΆ β (πΉ limβ π΅) β (πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π΅ β β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limc 25607 | . . 3 β’ limβ = (π β (β βpm β), π₯ β β β¦ {π¦ β£ [(TopOpenββfld) / π](π§ β (dom π βͺ {π₯}) β¦ if(π§ = π₯, π¦, (πβπ§))) β (((π βΎt (dom π βͺ {π₯})) CnP π)βπ₯)}) | |
2 | 1 | elmpocl 7650 | . 2 β’ (πΆ β (πΉ limβ π΅) β (πΉ β (β βpm β) β§ π΅ β β)) |
3 | cnex 11193 | . . . . 5 β’ β β V | |
4 | 3, 3 | elpm2 8870 | . . . 4 β’ (πΉ β (β βpm β) β (πΉ:dom πΉβΆβ β§ dom πΉ β β)) |
5 | 4 | anbi1i 624 | . . 3 β’ ((πΉ β (β βpm β) β§ π΅ β β) β ((πΉ:dom πΉβΆβ β§ dom πΉ β β) β§ π΅ β β)) |
6 | df-3an 1089 | . . 3 β’ ((πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π΅ β β) β ((πΉ:dom πΉβΆβ β§ dom πΉ β β) β§ π΅ β β)) | |
7 | 5, 6 | bitr4i 277 | . 2 β’ ((πΉ β (β βpm β) β§ π΅ β β) β (πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π΅ β β)) |
8 | 2, 7 | sylib 217 | 1 β’ (πΆ β (πΉ limβ π΅) β (πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π΅ β β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 β wcel 2106 {cab 2709 [wsbc 3777 βͺ cun 3946 β wss 3948 ifcif 4528 {csn 4628 β¦ cmpt 5231 dom cdm 5676 βΆwf 6539 βcfv 6543 (class class class)co 7411 βpm cpm 8823 βcc 11110 βΎt crest 17370 TopOpenctopn 17371 βfldccnfld 21144 CnP ccnp 22949 limβ climc 25603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-pm 8825 df-limc 25607 |
This theorem is referenced by: limccl 25616 limcdif 25617 limcresi 25626 limcres 25627 limccnp 25632 limccnp2 25633 limcco 25634 limcun 25636 mullimc 44631 limccog 44635 mullimcf 44638 limcperiod 44643 limcmptdm 44650 neglimc 44662 addlimc 44663 0ellimcdiv 44664 reclimc 44668 |
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