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| Mirrors > Home > ILE Home > Th. List > istopon | Unicode version | ||
| Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
| Ref | Expression |
|---|---|
| istopon |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funtopon 14484 |
. . . . 5
| |
| 2 | funrel 5288 |
. . . . 5
| |
| 3 | 1, 2 | ax-mp 5 |
. . . 4
|
| 4 | relelfvdm 5608 |
. . . 4
| |
| 5 | 3, 4 | mpan 424 |
. . 3
|
| 6 | 5 | elexd 2785 |
. 2
|
| 7 | uniexg 4486 |
. . . 4
| |
| 8 | eleq1 2268 |
. . . 4
| |
| 9 | 7, 8 | syl5ibrcom 157 |
. . 3
|
| 10 | 9 | imp 124 |
. 2
|
| 11 | eqeq1 2212 |
. . . . . 6
| |
| 12 | 11 | rabbidv 2761 |
. . . . 5
|
| 13 | df-topon 14483 |
. . . . 5
| |
| 14 | vpwex 4223 |
. . . . . . 7
| |
| 15 | 14 | pwex 4227 |
. . . . . 6
|
| 16 | rabss 3270 |
. . . . . . 7
| |
| 17 | pwuni 4236 |
. . . . . . . . . 10
| |
| 18 | pweq 3619 |
. . . . . . . . . 10
| |
| 19 | 17, 18 | sseqtrrid 3244 |
. . . . . . . . 9
|
| 20 | velpw 3623 |
. . . . . . . . 9
| |
| 21 | 19, 20 | sylibr 134 |
. . . . . . . 8
|
| 22 | 21 | a1i 9 |
. . . . . . 7
|
| 23 | 16, 22 | mprgbir 2564 |
. . . . . 6
|
| 24 | 15, 23 | ssexi 4182 |
. . . . 5
|
| 25 | 12, 13, 24 | fvmpt3i 5659 |
. . . 4
|
| 26 | 25 | eleq2d 2275 |
. . 3
|
| 27 | unieq 3859 |
. . . . 5
| |
| 28 | 27 | eqeq2d 2217 |
. . . 4
|
| 29 | 28 | elrab 2929 |
. . 3
|
| 30 | 26, 29 | bitrdi 196 |
. 2
|
| 31 | 6, 10, 30 | pm5.21nii 706 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-topon 14483 |
| This theorem is referenced by: topontop 14486 toponuni 14487 toptopon 14490 toponcom 14499 istps2 14505 tgtopon 14538 distopon 14559 epttop 14562 resttopon 14643 resttopon2 14650 txtopon 14734 |
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