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Mirrors > Home > ILE Home > Th. List > tg2 | Unicode version |
Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
Ref | Expression |
---|---|
tg2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topgen 12144 | . . . . . 6 | |
2 | 1 | funmpt2 5162 | . . . . 5 |
3 | funrel 5140 | . . . . 5 | |
4 | 2, 3 | ax-mp 5 | . . . 4 |
5 | relelfvdm 5453 | . . . 4 | |
6 | 4, 5 | mpan 420 | . . 3 |
7 | eltg2b 12226 | . . . 4 | |
8 | eleq1 2202 | . . . . . . 7 | |
9 | 8 | anbi1d 460 | . . . . . 6 |
10 | 9 | rexbidv 2438 | . . . . 5 |
11 | 10 | rspccv 2786 | . . . 4 |
12 | 7, 11 | syl6bi 162 | . . 3 |
13 | 6, 12 | mpcom 36 | . 2 |
14 | 13 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 cdm 4539 wrel 4544 wfun 5117 cfv 5123 ctg 12138 |
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 |
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-v 2688 df-sbc 2910 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-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 12144 |
This theorem is referenced by: tgclb 12237 tgcnp 12381 txlm 12451 |
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