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Mirrors > Home > ILE Home > Th. List > cnco | Unicode version |
Description: The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnco |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntop1 12281 | . . 3 | |
2 | cntop2 12282 | . . 3 | |
3 | 1, 2 | anim12i 336 | . 2 |
4 | eqid 2117 | . . . . 5 | |
5 | eqid 2117 | . . . . 5 | |
6 | 4, 5 | cnf 12284 | . . . 4 |
7 | eqid 2117 | . . . . 5 | |
8 | 7, 4 | cnf 12284 | . . . 4 |
9 | fco 5258 | . . . 4 | |
10 | 6, 8, 9 | syl2anr 288 | . . 3 |
11 | cnvco 4694 | . . . . . . 7 | |
12 | 11 | imaeq1i 4848 | . . . . . 6 |
13 | imaco 5014 | . . . . . 6 | |
14 | 12, 13 | eqtri 2138 | . . . . 5 |
15 | simpll 503 | . . . . . 6 | |
16 | cnima 12300 | . . . . . . 7 | |
17 | 16 | adantll 467 | . . . . . 6 |
18 | cnima 12300 | . . . . . 6 | |
19 | 15, 17, 18 | syl2anc 408 | . . . . 5 |
20 | 14, 19 | eqeltrid 2204 | . . . 4 |
21 | 20 | ralrimiva 2482 | . . 3 |
22 | 10, 21 | jca 304 | . 2 |
23 | 7, 5 | iscn2 12280 | . 2 |
24 | 3, 22, 23 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 wral 2393 cuni 3706 ccnv 4508 cima 4512 ccom 4513 wf 5089 (class class class)co 5742 ctop 12075 ccn 12265 |
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-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-top 12076 df-topon 12089 df-cn 12268 |
This theorem is referenced by: txcn 12355 cnmpt11 12363 cnmpt21 12371 hmeoco 12396 |
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