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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 12130 | . . . . . 6 | |
2 | 1 | funmpt2 5157 | . . . . 5 |
3 | funrel 5135 | . . . . 5 | |
4 | 2, 3 | ax-mp 5 | . . . 4 |
5 | relelfvdm 5446 | . . . 4 | |
6 | 4, 5 | mpan 420 | . . 3 |
7 | eltg2b 12212 | . . . 4 | |
8 | eleq1 2200 | . . . . . . 7 | |
9 | 8 | anbi1d 460 | . . . . . 6 |
10 | 9 | rexbidv 2436 | . . . . 5 |
11 | 10 | rspccv 2781 | . . . 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 2123 wral 2414 wrex 2415 cvv 2681 cin 3065 wss 3066 cpw 3505 cuni 3731 cdm 4534 wrel 4539 wfun 5112 cfv 5118 ctg 12124 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-topgen 12130 |
This theorem is referenced by: tgclb 12223 tgcnp 12367 txlm 12437 |
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