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Mirrors > Home > ILE Home > Th. List > tgclb | Unicode version |
Description: The property tgcl 12506 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 12506 | . 2 | |
2 | df-topgen 12414 | . . . . . . . . . . . . 13 | |
3 | 2 | funmpt2 5210 | . . . . . . . . . . . 12 |
4 | funrel 5188 | . . . . . . . . . . . 12 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . 11 |
6 | 0opn 12446 | . . . . . . . . . . 11 | |
7 | relelfvdm 5501 | . . . . . . . . . . 11 | |
8 | 5, 6, 7 | sylancr 411 | . . . . . . . . . 10 |
9 | 8 | elexd 2725 | . . . . . . . . 9 |
10 | bastg 12503 | . . . . . . . . 9 | |
11 | 9, 10 | syl 14 | . . . . . . . 8 |
12 | 11 | sselda 3128 | . . . . . . 7 |
13 | 11 | sselda 3128 | . . . . . . 7 |
14 | 12, 13 | anim12dan 590 | . . . . . 6 |
15 | inopn 12443 | . . . . . . 7 | |
16 | 15 | 3expb 1186 | . . . . . 6 |
17 | 14, 16 | syldan 280 | . . . . 5 |
18 | tg2 12502 | . . . . . 6 | |
19 | 18 | ralrimiva 2530 | . . . . 5 |
20 | 17, 19 | syl 14 | . . . 4 |
21 | 20 | ralrimivva 2539 | . . 3 |
22 | isbasis2g 12485 | . . . 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 2128 cab 2143 wral 2435 wrex 2436 cvv 2712 cin 3101 wss 3102 c0 3394 cpw 3543 cuni 3773 cdm 4587 wrel 4592 wfun 5165 cfv 5171 ctg 12408 ctop 12437 ctb 12482 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-topgen 12414 df-top 12438 df-bases 12483 |
This theorem is referenced by: bastop2 12526 tgcn 12650 tgcnp 12651 |
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