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Mirrors > Home > ILE Home > Th. List > limcresi | Unicode version |
Description: Any limit of is also a limit of the restriction of . (Contributed by Mario Carneiro, 28-Dec-2016.) |
Ref | Expression |
---|---|
limcresi | lim lim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcrcl 12785 | . . . . . . 7 lim | |
2 | 1 | simp1d 993 | . . . . . 6 lim |
3 | 1 | simp2d 994 | . . . . . 6 lim |
4 | 1 | simp3d 995 | . . . . . 6 lim |
5 | 2, 3, 4 | ellimc3ap 12788 | . . . . 5 lim lim # |
6 | 5 | ibi 175 | . . . 4 lim # |
7 | inss1 3291 | . . . . . . . . 9 | |
8 | ssralv 3156 | . . . . . . . . 9 # # | |
9 | 7, 8 | ax-mp 5 | . . . . . . . 8 # # |
10 | elinel2 3258 | . . . . . . . . . . . . . . 15 | |
11 | fvres 5438 | . . . . . . . . . . . . . . 15 | |
12 | 10, 11 | syl 14 | . . . . . . . . . . . . . 14 |
13 | 12 | adantl 275 | . . . . . . . . . . . . 13 lim |
14 | 13 | fvoveq1d 5789 | . . . . . . . . . . . 12 lim |
15 | 14 | breq1d 3934 | . . . . . . . . . . 11 lim |
16 | 15 | imbi2d 229 | . . . . . . . . . 10 lim # # |
17 | 16 | biimprd 157 | . . . . . . . . 9 lim # # |
18 | 17 | ralimdva 2497 | . . . . . . . 8 lim # # |
19 | 9, 18 | syl5 32 | . . . . . . 7 lim # # |
20 | 19 | reximdv 2531 | . . . . . 6 lim # # |
21 | 20 | ralimdv 2498 | . . . . 5 lim # # |
22 | 21 | anim2d 335 | . . . 4 lim # # |
23 | 6, 22 | mpd 13 | . . 3 lim # |
24 | fresin 5296 | . . . . 5 | |
25 | 2, 24 | syl 14 | . . . 4 lim |
26 | 7, 3 | sstrid 3103 | . . . 4 lim |
27 | 25, 26, 4 | ellimc3ap 12788 | . . 3 lim lim # |
28 | 23, 27 | mpbird 166 | . 2 lim lim |
29 | 28 | ssriv 3096 | 1 lim lim |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 wrex 2415 cin 3065 wss 3066 class class class wbr 3924 cdm 4534 cres 4536 wf 5114 cfv 5118 (class class class)co 5767 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-res 4546 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: dvidlemap 12818 dvcnp2cntop 12821 dvcoapbr 12829 |
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