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Mirrors > Home > ILE Home > Th. List > tgclb | Unicode version |
Description: The property tgcl 12233 can be reversed: if the topology generated by is actually a topology, then must be a topological basis. This yields an alternative definition of . (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgclb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcl 12233 | . 2 | |
2 | df-topgen 12141 | . . . . . . . . . . . . 13 | |
3 | 2 | funmpt2 5162 | . . . . . . . . . . . 12 |
4 | funrel 5140 | . . . . . . . . . . . 12 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . 11 |
6 | 0opn 12173 | . . . . . . . . . . 11 | |
7 | relelfvdm 5453 | . . . . . . . . . . 11 | |
8 | 5, 6, 7 | sylancr 410 | . . . . . . . . . 10 |
9 | 8 | elexd 2699 | . . . . . . . . 9 |
10 | bastg 12230 | . . . . . . . . 9 | |
11 | 9, 10 | syl 14 | . . . . . . . 8 |
12 | 11 | sselda 3097 | . . . . . . 7 |
13 | 11 | sselda 3097 | . . . . . . 7 |
14 | 12, 13 | anim12dan 589 | . . . . . 6 |
15 | inopn 12170 | . . . . . . 7 | |
16 | 15 | 3expb 1182 | . . . . . 6 |
17 | 14, 16 | syldan 280 | . . . . 5 |
18 | tg2 12229 | . . . . . 6 | |
19 | 18 | ralrimiva 2505 | . . . . 5 |
20 | 17, 19 | syl 14 | . . . 4 |
21 | 20 | ralrimivva 2514 | . . 3 |
22 | isbasis2g 12212 | . . . 4 | |
23 | 9, 22 | syl 14 | . . 3 |
24 | 21, 23 | mpbird 166 | . 2 |
25 | 1, 24 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 cin 3070 wss 3071 c0 3363 cpw 3510 cuni 3736 cdm 4539 wrel 4544 wfun 5117 cfv 5123 ctg 12135 ctop 12164 ctb 12209 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 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-iota 5088 df-fun 5125 df-fv 5131 df-topgen 12141 df-top 12165 df-bases 12210 |
This theorem is referenced by: bastop2 12253 tgcn 12377 tgcnp 12378 |
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