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Mirrors > Home > ILE Home > Th. List > cnmpt22f | Unicode version |
Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmpt21.j | TopOn |
cnmpt21.k | TopOn |
cnmpt21.a | |
cnmpt2t.b | |
cnmpt22f.f |
Ref | Expression |
---|---|
cnmpt22f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt21.j | . 2 TopOn | |
2 | cnmpt21.k | . 2 TopOn | |
3 | cnmpt21.a | . 2 | |
4 | cnmpt2t.b | . 2 | |
5 | cntop2 12381 | . . . 4 | |
6 | 3, 5 | syl 14 | . . 3 |
7 | toptopon2 12196 | . . 3 TopOn | |
8 | 6, 7 | sylib 121 | . 2 TopOn |
9 | cntop2 12381 | . . . 4 | |
10 | 4, 9 | syl 14 | . . 3 |
11 | toptopon2 12196 | . . 3 TopOn | |
12 | 10, 11 | sylib 121 | . 2 TopOn |
13 | txtopon 12441 | . . . . . . 7 TopOn TopOn TopOn | |
14 | 8, 12, 13 | syl2anc 408 | . . . . . 6 TopOn |
15 | cnmpt22f.f | . . . . . . . 8 | |
16 | cntop2 12381 | . . . . . . . 8 | |
17 | 15, 16 | syl 14 | . . . . . . 7 |
18 | toptopon2 12196 | . . . . . . 7 TopOn | |
19 | 17, 18 | sylib 121 | . . . . . 6 TopOn |
20 | cnf2 12384 | . . . . . 6 TopOn TopOn | |
21 | 14, 19, 15, 20 | syl3anc 1216 | . . . . 5 |
22 | 21 | ffnd 5273 | . . . 4 |
23 | fnovim 5879 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | 24, 15 | eqeltrrd 2217 | . 2 |
26 | oveq12 5783 | . 2 | |
27 | 1, 2, 3, 4, 8, 12, 25, 26 | cnmpt22 12473 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 cuni 3736 cxp 4537 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 ctop 12174 TopOnctopon 12187 ccn 12364 ctx 12431 |
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 12151 df-top 12175 df-topon 12188 df-bases 12220 df-cn 12367 df-tx 12432 |
This theorem is referenced by: cnmptcom 12477 divcnap 12734 cnrehmeocntop 12772 |
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