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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 13241. (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 13243 | . . . 4 TopOn | |
3 | 1, 2 | syl 14 | . . 3 |
4 | 3 | breqd 4009 | . 2 |
5 | reseq1 4894 | . . . . . . . . 9 | |
6 | 5 | feq1d 5344 | . . . . . . . 8 |
7 | 6 | rexbidv 2476 | . . . . . . 7 |
8 | 7 | imbi2d 230 | . . . . . 6 |
9 | 8 | ralbidv 2475 | . . . . 5 |
10 | eleq1 2238 | . . . . . . 7 | |
11 | 10 | imbi1d 231 | . . . . . 6 |
12 | 11 | ralbidv 2475 | . . . . 5 |
13 | 9, 12 | sylan9bb 462 | . . . 4 |
14 | df-3an 980 | . . . . 5 | |
15 | 14 | opabbii 4065 | . . . 4 |
16 | 13, 15 | brab2a 4673 | . . 3 |
17 | df-3an 980 | . . 3 | |
18 | 16, 17 | bitr4i 187 | . 2 |
19 | 4, 18 | bitrdi 196 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 wral 2453 wrex 2454 class class class wbr 3998 copab 4058 crn 4621 cres 4622 wf 5204 cfv 5208 (class class class)co 5865 cpm 6639 cc 7784 cuz 9499 TopOnctopon 13059 clm 13238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-cnex 7877 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-pm 6641 df-top 13047 df-topon 13060 df-lm 13241 |
This theorem is referenced by: lmbr2 13265 lmfpm 13294 lmcl 13296 lmff 13300 |
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