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Mirrors > Home > ILE Home > Th. List > txswaphmeo | Unicode version |
Description: There is a homeomorphism from to . (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
txswaphmeo | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 TopOn TopOn TopOn | |
2 | simpr 109 | . . 3 TopOn TopOn TopOn | |
3 | 1, 2 | cnmpt2nd 12461 | . . 3 TopOn TopOn |
4 | 1, 2 | cnmpt1st 12460 | . . 3 TopOn TopOn |
5 | 1, 2, 3, 4 | cnmpt2t 12465 | . 2 TopOn TopOn |
6 | opelxpi 4571 | . . . . . . . . 9 | |
7 | 6 | ancoms 266 | . . . . . . . 8 |
8 | 7 | adantl 275 | . . . . . . 7 TopOn TopOn |
9 | 8 | ralrimivva 2514 | . . . . . 6 TopOn TopOn |
10 | eqid 2139 | . . . . . . 7 | |
11 | 10 | fmpo 6099 | . . . . . 6 |
12 | 9, 11 | sylib 121 | . . . . 5 TopOn TopOn |
13 | opelxpi 4571 | . . . . . . . . 9 | |
14 | 13 | ancoms 266 | . . . . . . . 8 |
15 | 14 | adantl 275 | . . . . . . 7 TopOn TopOn |
16 | 15 | ralrimivva 2514 | . . . . . 6 TopOn TopOn |
17 | eqid 2139 | . . . . . . 7 | |
18 | 17 | fmpo 6099 | . . . . . 6 |
19 | 16, 18 | sylib 121 | . . . . 5 TopOn TopOn |
20 | txswaphmeolem 12492 | . . . . . 6 | |
21 | txswaphmeolem 12492 | . . . . . 6 | |
22 | fcof1o 5690 | . . . . . 6 | |
23 | 20, 21, 22 | mpanr12 435 | . . . . 5 |
24 | 12, 19, 23 | syl2anc 408 | . . . 4 TopOn TopOn |
25 | 24 | simprd 113 | . . 3 TopOn TopOn |
26 | 2, 1 | cnmpt2nd 12461 | . . . 4 TopOn TopOn |
27 | 2, 1 | cnmpt1st 12460 | . . . 4 TopOn TopOn |
28 | 2, 1, 26, 27 | cnmpt2t 12465 | . . 3 TopOn TopOn |
29 | 25, 28 | eqeltrd 2216 | . 2 TopOn TopOn |
30 | ishmeo 12476 | . 2 | |
31 | 5, 29, 30 | sylanbrc 413 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cop 3530 cid 4210 cxp 4537 ccnv 4538 cres 4541 ccom 4543 wf 5119 wf1o 5122 cfv 5123 (class class class)co 5774 cmpo 5776 TopOnctopon 12180 ccn 12357 ctx 12424 chmeo 12472 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-topgen 12144 df-top 12168 df-topon 12181 df-bases 12213 df-cn 12360 df-tx 12425 df-hmeo 12473 |
This theorem is referenced by: (None) |
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