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Mirrors > Home > ILE Home > Th. List > ssidcn | Unicode version |
Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
ssidcn | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscn 12293 | . . 3 TopOn TopOn | |
2 | f1oi 5373 | . . . . 5 | |
3 | f1of 5335 | . . . . 5 | |
4 | 2, 3 | ax-mp 5 | . . . 4 |
5 | 4 | biantrur 301 | . . 3 |
6 | 1, 5 | syl6bbr 197 | . 2 TopOn TopOn |
7 | cnvresid 5167 | . . . . . . 7 | |
8 | 7 | imaeq1i 4848 | . . . . . 6 |
9 | elssuni 3734 | . . . . . . . . 9 | |
10 | 9 | adantl 275 | . . . . . . . 8 TopOn TopOn |
11 | toponuni 12109 | . . . . . . . . 9 TopOn | |
12 | 11 | ad2antlr 480 | . . . . . . . 8 TopOn TopOn |
13 | 10, 12 | sseqtrrd 3106 | . . . . . . 7 TopOn TopOn |
14 | resiima 4867 | . . . . . . 7 | |
15 | 13, 14 | syl 14 | . . . . . 6 TopOn TopOn |
16 | 8, 15 | syl5eq 2162 | . . . . 5 TopOn TopOn |
17 | 16 | eleq1d 2186 | . . . 4 TopOn TopOn |
18 | 17 | ralbidva 2410 | . . 3 TopOn TopOn |
19 | dfss3 3057 | . . 3 | |
20 | 18, 19 | syl6bbr 197 | . 2 TopOn TopOn |
21 | 6, 20 | bitrd 187 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 wss 3041 cuni 3706 cid 4180 ccnv 4508 cres 4511 cima 4512 wf 5089 wf1o 5092 cfv 5093 (class class class)co 5742 TopOnctopon 12104 ccn 12281 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-top 12092 df-topon 12105 df-cn 12284 |
This theorem is referenced by: idcn 12308 |
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