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Mirrors > Home > ILE Home > Th. List > limccl | Unicode version |
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
limccl | lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 lim lim | |
2 | df-limced 12783 | . . . . . 6 lim # | |
3 | 2 | elmpocl1 5962 | . . . . 5 lim |
4 | limcrcl 12785 | . . . . . 6 lim | |
5 | 4 | simp3d 995 | . . . . 5 lim |
6 | cnex 7737 | . . . . . . 7 | |
7 | 6 | rabex 4067 | . . . . . 6 # |
8 | 7 | a1i 9 | . . . . 5 lim # |
9 | simpl 108 | . . . . . . . . . 10 | |
10 | 9 | dmeqd 4736 | . . . . . . . . . 10 |
11 | 9, 10 | feq12d 5257 | . . . . . . . . 9 |
12 | 10 | sseq1d 3121 | . . . . . . . . 9 |
13 | 11, 12 | anbi12d 464 | . . . . . . . 8 |
14 | simpr 109 | . . . . . . . . . 10 | |
15 | 14 | eleq1d 2206 | . . . . . . . . 9 |
16 | 14 | breq2d 3936 | . . . . . . . . . . . . . 14 # # |
17 | 14 | oveq2d 5783 | . . . . . . . . . . . . . . . 16 |
18 | 17 | fveq2d 5418 | . . . . . . . . . . . . . . 15 |
19 | 18 | breq1d 3934 | . . . . . . . . . . . . . 14 |
20 | 16, 19 | anbi12d 464 | . . . . . . . . . . . . 13 # # |
21 | 9 | fveq1d 5416 | . . . . . . . . . . . . . . 15 |
22 | 21 | fvoveq1d 5789 | . . . . . . . . . . . . . 14 |
23 | 22 | breq1d 3934 | . . . . . . . . . . . . 13 |
24 | 20, 23 | imbi12d 233 | . . . . . . . . . . . 12 # # |
25 | 10, 24 | raleqbidv 2636 | . . . . . . . . . . 11 # # |
26 | 25 | rexbidv 2436 | . . . . . . . . . 10 # # |
27 | 26 | ralbidv 2435 | . . . . . . . . 9 # # |
28 | 15, 27 | anbi12d 464 | . . . . . . . 8 # # |
29 | 13, 28 | anbi12d 464 | . . . . . . 7 # # |
30 | 29 | rabbidv 2670 | . . . . . 6 # # |
31 | 30, 2 | ovmpoga 5893 | . . . . 5 # lim # |
32 | 3, 5, 8, 31 | syl3anc 1216 | . . . 4 lim lim # |
33 | 1, 32 | eleqtrd 2216 | . . 3 lim # |
34 | elrabi 2832 | . . 3 # | |
35 | 33, 34 | syl 14 | . 2 lim |
36 | 35 | ssriv 3096 | 1 lim |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 wrex 2415 crab 2418 cvv 2681 wss 3066 class class class wbr 3924 cdm 4534 wf 5114 cfv 5118 (class class class)co 5767 cpm 6536 cc 7611 clt 7793 cmin 7926 # cap 8336 crp 9434 cabs 10762 lim climc 12781 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pm 6538 df-limced 12783 |
This theorem is referenced by: reldvg 12806 dvfvalap 12808 dvcl 12810 |
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