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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 12737. (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 12739 | . . . 4 TopOn | |
3 | 1, 2 | syl 14 | . . 3 |
4 | 3 | breqd 3987 | . 2 |
5 | reseq1 4872 | . . . . . . . . 9 | |
6 | 5 | feq1d 5318 | . . . . . . . 8 |
7 | 6 | rexbidv 2465 | . . . . . . 7 |
8 | 7 | imbi2d 229 | . . . . . 6 |
9 | 8 | ralbidv 2464 | . . . . 5 |
10 | eleq1 2227 | . . . . . . 7 | |
11 | 10 | imbi1d 230 | . . . . . 6 |
12 | 11 | ralbidv 2464 | . . . . 5 |
13 | 9, 12 | sylan9bb 458 | . . . 4 |
14 | df-3an 969 | . . . . 5 | |
15 | 14 | opabbii 4043 | . . . 4 |
16 | 13, 15 | brab2a 4651 | . . 3 |
17 | df-3an 969 | . . 3 | |
18 | 16, 17 | bitr4i 186 | . 2 |
19 | 4, 18 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 class class class wbr 3976 copab 4036 crn 4599 cres 4600 wf 5178 cfv 5182 (class class class)co 5836 cpm 6606 cc 7742 cuz 9457 TopOnctopon 12555 clm 12734 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-cnex 7835 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pm 6608 df-top 12543 df-topon 12556 df-lm 12737 |
This theorem is referenced by: lmbr2 12761 lmfpm 12790 lmcl 12792 lmff 12796 |
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