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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 14486 |
. . . 4
| |
| 2 | 1 | adantr 276 |
. . 3
|
| 3 | id 19 |
. . . 4
| |
| 4 | toponmax 14497 |
. . . 4
| |
| 5 | ssexg 4183 |
. . . 4
| |
| 6 | 3, 4, 5 | syl2anr 290 |
. . 3
|
| 7 | resttop 14642 |
. . 3
| |
| 8 | 2, 6, 7 | syl2anc 411 |
. 2
|
| 9 | simpr 110 |
. . . . . 6
| |
| 10 | sseqin2 3392 |
. . . . . 6
| |
| 11 | 9, 10 | sylib 122 |
. . . . 5
|
| 12 | simpl 109 |
. . . . . 6
| |
| 13 | 4 | adantr 276 |
. . . . . 6
|
| 14 | elrestr 13079 |
. . . . . 6
| |
| 15 | 12, 6, 13, 14 | syl3anc 1250 |
. . . . 5
|
| 16 | 11, 15 | eqeltrrd 2283 |
. . . 4
|
| 17 | elssuni 3878 |
. . . 4
| |
| 18 | 16, 17 | syl 14 |
. . 3
|
| 19 | restval 13077 |
. . . . . 6
| |
| 20 | 6, 19 | syldan 282 |
. . . . 5
|
| 21 | inss2 3394 |
. . . . . . . . 9
| |
| 22 | vex 2775 |
. . . . . . . . . . 11
| |
| 23 | 22 | inex1 4178 |
. . . . . . . . . 10
|
| 24 | 23 | elpw 3622 |
. . . . . . . . 9
|
| 25 | 21, 24 | mpbir 146 |
. . . . . . . 8
|
| 26 | 25 | a1i 9 |
. . . . . . 7
|
| 27 | 26 | fmpttd 5735 |
. . . . . 6
|
| 28 | 27 | frnd 5435 |
. . . . 5
|
| 29 | 20, 28 | eqsstrd 3229 |
. . . 4
|
| 30 | sspwuni 4012 |
. . . 4
| |
| 31 | 29, 30 | sylib 122 |
. . 3
|
| 32 | 18, 31 | eqssd 3210 |
. 2
|
| 33 | istopon 14485 |
. 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 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-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 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-ne 2377 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 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-iun 3929 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-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-rest 13073 df-topgen 13092 df-top 14470 df-topon 14483 df-bases 14515 |
| This theorem is referenced by: restuni 14644 stoig 14645 cnrest 14707 cnrest2 14708 cnrest2r 14709 cnptopresti 14710 cnptoprest 14711 cnptoprest2 14712 divcnap 15037 cncfmpt2fcntop 15071 cnplimcim 15139 cnlimcim 15143 cnlimc 15144 limccnpcntop 15147 limccnp2lem 15148 limccnp2cntop 15149 dvfvalap 15153 dvbss 15157 dvfgg 15160 dvcnp2cntop 15171 dvcn 15172 dvaddxxbr 15173 dvmulxxbr 15174 dvmptfsum 15197 |
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