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| Mirrors > Home > ILE Home > Th. List > resttopon | Unicode version | ||
| Description: A subspace topology is a topology on the base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
| Ref | Expression |
|---|---|
| resttopon |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | topontop 14457 |
. . . 4
| |
| 2 | 1 | adantr 276 |
. . 3
|
| 3 | id 19 |
. . . 4
| |
| 4 | toponmax 14468 |
. . . 4
| |
| 5 | ssexg 4182 |
. . . 4
| |
| 6 | 3, 4, 5 | syl2anr 290 |
. . 3
|
| 7 | resttop 14613 |
. . 3
| |
| 8 | 2, 6, 7 | syl2anc 411 |
. 2
|
| 9 | simpr 110 |
. . . . . 6
| |
| 10 | sseqin2 3391 |
. . . . . 6
| |
| 11 | 9, 10 | sylib 122 |
. . . . 5
|
| 12 | simpl 109 |
. . . . . 6
| |
| 13 | 4 | adantr 276 |
. . . . . 6
|
| 14 | elrestr 13050 |
. . . . . 6
| |
| 15 | 12, 6, 13, 14 | syl3anc 1249 |
. . . . 5
|
| 16 | 11, 15 | eqeltrrd 2282 |
. . . 4
|
| 17 | elssuni 3877 |
. . . 4
| |
| 18 | 16, 17 | syl 14 |
. . 3
|
| 19 | restval 13048 |
. . . . . 6
| |
| 20 | 6, 19 | syldan 282 |
. . . . 5
|
| 21 | inss2 3393 |
. . . . . . . . 9
| |
| 22 | vex 2774 |
. . . . . . . . . . 11
| |
| 23 | 22 | inex1 4177 |
. . . . . . . . . 10
|
| 24 | 23 | elpw 3621 |
. . . . . . . . 9
|
| 25 | 21, 24 | mpbir 146 |
. . . . . . . 8
|
| 26 | 25 | a1i 9 |
. . . . . . 7
|
| 27 | 26 | fmpttd 5734 |
. . . . . 6
|
| 28 | 27 | frnd 5434 |
. . . . 5
|
| 29 | 20, 28 | eqsstrd 3228 |
. . . 4
|
| 30 | sspwuni 4011 |
. . . 4
| |
| 31 | 29, 30 | sylib 122 |
. . 3
|
| 32 | 18, 31 | eqssd 3209 |
. 2
|
| 33 | istopon 14456 |
. 2
| |
| 34 | 8, 32, 33 | sylanbrc 417 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-rest 13044 df-topgen 13063 df-top 14441 df-topon 14454 df-bases 14486 |
| This theorem is referenced by: restuni 14615 stoig 14616 cnrest 14678 cnrest2 14679 cnrest2r 14680 cnptopresti 14681 cnptoprest 14682 cnptoprest2 14683 divcnap 15008 cncfmpt2fcntop 15042 cnplimcim 15110 cnlimcim 15114 cnlimc 15115 limccnpcntop 15118 limccnp2lem 15119 limccnp2cntop 15120 dvfvalap 15124 dvbss 15128 dvfgg 15131 dvcnp2cntop 15142 dvcn 15143 dvaddxxbr 15144 dvmulxxbr 15145 dvmptfsum 15168 |
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