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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funtopon 13170 |
. . . . 5
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2 | funrel 5229 |
. . . . 5
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3 | 1, 2 | ax-mp 5 |
. . . 4
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4 | relelfvdm 5543 |
. . . 4
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5 | 3, 4 | mpan 424 |
. . 3
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6 | 5 | elexd 2750 |
. 2
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7 | uniexg 4436 |
. . . 4
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8 | eleq1 2240 |
. . . 4
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9 | 7, 8 | syl5ibrcom 157 |
. . 3
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10 | 9 | imp 124 |
. 2
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11 | eqeq1 2184 |
. . . . . 6
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12 | 11 | rabbidv 2726 |
. . . . 5
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13 | df-topon 13169 |
. . . . 5
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14 | vpwex 4176 |
. . . . . . 7
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15 | 14 | pwex 4180 |
. . . . . 6
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16 | rabss 3232 |
. . . . . . 7
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17 | pwuni 4189 |
. . . . . . . . . 10
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18 | pweq 3577 |
. . . . . . . . . 10
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19 | 17, 18 | sseqtrrid 3206 |
. . . . . . . . 9
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20 | velpw 3581 |
. . . . . . . . 9
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21 | 19, 20 | sylibr 134 |
. . . . . . . 8
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22 | 21 | a1i 9 |
. . . . . . 7
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23 | 16, 22 | mprgbir 2535 |
. . . . . 6
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24 | 15, 23 | ssexi 4138 |
. . . . 5
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25 | 12, 13, 24 | fvmpt3i 5592 |
. . . 4
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26 | 25 | eleq2d 2247 |
. . 3
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27 | unieq 3816 |
. . . . 5
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28 | 27 | eqeq2d 2189 |
. . . 4
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29 | 28 | elrab 2893 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 26, 29 | bitrdi 196 |
. 2
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31 | 6, 10, 30 | pm5.21nii 704 |
1
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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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-topon 13169 |
This theorem is referenced by: topontop 13172 toponuni 13173 toptopon 13176 toponcom 13185 istps2 13191 tgtopon 13226 distopon 13247 epttop 13250 resttopon 13331 resttopon2 13338 txtopon 13422 |
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