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Mirrors > Home > ILE Home > Th. List > hmeontr | Unicode version |
Description: Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoopn.1 |
Ref | Expression |
---|---|
hmeontr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 12747 | . . . . . 6 | |
2 | 1 | adantr 274 | . . . . 5 |
3 | imassrn 4940 | . . . . . 6 | |
4 | hmeoopn.1 | . . . . . . . . 9 | |
5 | eqid 2157 | . . . . . . . . 9 | |
6 | 4, 5 | hmeof1o 12751 | . . . . . . . 8 |
7 | 6 | adantr 274 | . . . . . . 7 |
8 | f1ofo 5422 | . . . . . . 7 | |
9 | forn 5396 | . . . . . . 7 | |
10 | 7, 8, 9 | 3syl 17 | . . . . . 6 |
11 | 3, 10 | sseqtrid 3178 | . . . . 5 |
12 | 5 | cnntri 12666 | . . . . 5 |
13 | 2, 11, 12 | syl2anc 409 | . . . 4 |
14 | f1of1 5414 | . . . . . . 7 | |
15 | 7, 14 | syl 14 | . . . . . 6 |
16 | f1imacnv 5432 | . . . . . 6 | |
17 | 15, 16 | sylancom 417 | . . . . 5 |
18 | 17 | fveq2d 5473 | . . . 4 |
19 | 13, 18 | sseqtrd 3166 | . . 3 |
20 | f1ofun 5417 | . . . . 5 | |
21 | 7, 20 | syl 14 | . . . 4 |
22 | cntop2 12644 | . . . . . . 7 | |
23 | 2, 22 | syl 14 | . . . . . 6 |
24 | 5 | ntrss3 12565 | . . . . . 6 |
25 | 23, 11, 24 | syl2anc 409 | . . . . 5 |
26 | 25, 10 | sseqtrrd 3167 | . . . 4 |
27 | funimass1 5248 | . . . 4 | |
28 | 21, 26, 27 | syl2anc 409 | . . 3 |
29 | 19, 28 | mpd 13 | . 2 |
30 | hmeocnvcn 12748 | . . . 4 | |
31 | 4 | cnntri 12666 | . . . 4 |
32 | 30, 31 | sylan 281 | . . 3 |
33 | imacnvcnv 5051 | . . 3 | |
34 | imacnvcnv 5051 | . . . 4 | |
35 | 34 | fveq2i 5472 | . . 3 |
36 | 32, 33, 35 | 3sstr3g 3170 | . 2 |
37 | 29, 36 | eqssd 3145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wss 3102 cuni 3773 ccnv 4586 crn 4588 cima 4590 wfun 5165 wf1 5168 wfo 5169 wf1o 5170 cfv 5171 (class class class)co 5825 ctop 12437 cnt 12535 ccn 12627 chmeo 12742 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-map 6596 df-top 12438 df-topon 12451 df-ntr 12538 df-cn 12630 df-hmeo 12743 |
This theorem is referenced by: (None) |
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