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Mirrors > Home > ILE Home > Th. List > lmbr | Unicode version |
Description: Express the binary relation "sequence converges to point " in a topological space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 12359. (Contributed by Mario Carneiro, 14-Nov-2013.) |
Ref | Expression |
---|---|
lmbr.2 | TopOn |
Ref | Expression |
---|---|
lmbr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmbr.2 | . . . 4 TopOn | |
2 | lmfval 12361 | . . . 4 TopOn | |
3 | 1, 2 | syl 14 | . . 3 |
4 | 3 | breqd 3940 | . 2 |
5 | reseq1 4813 | . . . . . . . . 9 | |
6 | 5 | feq1d 5259 | . . . . . . . 8 |
7 | 6 | rexbidv 2438 | . . . . . . 7 |
8 | 7 | imbi2d 229 | . . . . . 6 |
9 | 8 | ralbidv 2437 | . . . . 5 |
10 | eleq1 2202 | . . . . . . 7 | |
11 | 10 | imbi1d 230 | . . . . . 6 |
12 | 11 | ralbidv 2437 | . . . . 5 |
13 | 9, 12 | sylan9bb 457 | . . . 4 |
14 | df-3an 964 | . . . . 5 | |
15 | 14 | opabbii 3995 | . . . 4 |
16 | 13, 15 | brab2a 4592 | . . 3 |
17 | df-3an 964 | . . 3 | |
18 | 16, 17 | bitr4i 186 | . 2 |
19 | 4, 18 | syl6bb 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 class class class wbr 3929 copab 3988 crn 4540 cres 4541 wf 5119 cfv 5123 (class class class)co 5774 cpm 6543 cc 7618 cuz 9326 TopOnctopon 12177 clm 12356 |
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-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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-cnex 7711 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pm 6545 df-top 12165 df-topon 12178 df-lm 12359 |
This theorem is referenced by: lmbr2 12383 lmfpm 12412 lmcl 12414 lmff 12418 |
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