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Mirrors > Home > ILE Home > Th. List > cnmpt22 | 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 | |
cnmpt22.l | TopOn |
cnmpt22.m | TopOn |
cnmpt22.c | |
cnmpt22.d |
Ref | Expression |
---|---|
cnmpt22 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5777 | . . . 4 | |
2 | cnmpt21.j | . . . . . . . . . 10 TopOn | |
3 | cnmpt21.k | . . . . . . . . . 10 TopOn | |
4 | txtopon 12431 | . . . . . . . . . 10 TopOn TopOn TopOn | |
5 | 2, 3, 4 | syl2anc 408 | . . . . . . . . 9 TopOn |
6 | cnmpt22.l | . . . . . . . . 9 TopOn | |
7 | cnmpt21.a | . . . . . . . . 9 | |
8 | cnf2 12374 | . . . . . . . . 9 TopOn TopOn | |
9 | 5, 6, 7, 8 | syl3anc 1216 | . . . . . . . 8 |
10 | eqid 2139 | . . . . . . . . 9 | |
11 | 10 | fmpo 6099 | . . . . . . . 8 |
12 | 9, 11 | sylibr 133 | . . . . . . 7 |
13 | rsp2 2482 | . . . . . . 7 | |
14 | 12, 13 | syl 14 | . . . . . 6 |
15 | 14 | 3impib 1179 | . . . . 5 |
16 | cnmpt22.m | . . . . . . . . 9 TopOn | |
17 | cnmpt2t.b | . . . . . . . . 9 | |
18 | cnf2 12374 | . . . . . . . . 9 TopOn TopOn | |
19 | 5, 16, 17, 18 | syl3anc 1216 | . . . . . . . 8 |
20 | eqid 2139 | . . . . . . . . 9 | |
21 | 20 | fmpo 6099 | . . . . . . . 8 |
22 | 19, 21 | sylibr 133 | . . . . . . 7 |
23 | rsp2 2482 | . . . . . . 7 | |
24 | 22, 23 | syl 14 | . . . . . 6 |
25 | 24 | 3impib 1179 | . . . . 5 |
26 | 15, 25 | jca 304 | . . . . . 6 |
27 | txtopon 12431 | . . . . . . . . . . 11 TopOn TopOn TopOn | |
28 | 6, 16, 27 | syl2anc 408 | . . . . . . . . . 10 TopOn |
29 | cnmpt22.c | . . . . . . . . . . . 12 | |
30 | cntop2 12371 | . . . . . . . . . . . 12 | |
31 | 29, 30 | syl 14 | . . . . . . . . . . 11 |
32 | toptopon2 12186 | . . . . . . . . . . 11 TopOn | |
33 | 31, 32 | sylib 121 | . . . . . . . . . 10 TopOn |
34 | cnf2 12374 | . . . . . . . . . 10 TopOn TopOn | |
35 | 28, 33, 29, 34 | syl3anc 1216 | . . . . . . . . 9 |
36 | eqid 2139 | . . . . . . . . . 10 | |
37 | 36 | fmpo 6099 | . . . . . . . . 9 |
38 | 35, 37 | sylibr 133 | . . . . . . . 8 |
39 | r2al 2454 | . . . . . . . 8 | |
40 | 38, 39 | sylib 121 | . . . . . . 7 |
41 | 40 | 3ad2ant1 1002 | . . . . . 6 |
42 | eleq1 2202 | . . . . . . . . 9 | |
43 | eleq1 2202 | . . . . . . . . 9 | |
44 | 42, 43 | bi2anan9 595 | . . . . . . . 8 |
45 | cnmpt22.d | . . . . . . . . 9 | |
46 | 45 | eleq1d 2208 | . . . . . . . 8 |
47 | 44, 46 | imbi12d 233 | . . . . . . 7 |
48 | 47 | spc2gv 2776 | . . . . . 6 |
49 | 26, 41, 26, 48 | syl3c 63 | . . . . 5 |
50 | 45, 36 | ovmpoga 5900 | . . . . 5 |
51 | 15, 25, 49, 50 | syl3anc 1216 | . . . 4 |
52 | 1, 51 | syl5eqr 2186 | . . 3 |
53 | 52 | mpoeq3dva 5835 | . 2 |
54 | 2, 3, 7, 17 | cnmpt2t 12462 | . . 3 |
55 | 2, 3, 54, 29 | cnmpt21f 12461 | . 2 |
56 | 53, 55 | eqeltrrd 2217 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2416 cop 3530 cuni 3736 cxp 4537 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 ctop 12164 TopOnctopon 12177 ccn 12354 ctx 12421 |
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 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-tx 12422 |
This theorem is referenced by: cnmpt22f 12464 |
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