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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 5839 | . . . 4 | |
2 | cnmpt21.j | . . . . . . . . . 10 TopOn | |
3 | cnmpt21.k | . . . . . . . . . 10 TopOn | |
4 | txtopon 12809 | . . . . . . . . . 10 TopOn TopOn TopOn | |
5 | 2, 3, 4 | syl2anc 409 | . . . . . . . . 9 TopOn |
6 | cnmpt22.l | . . . . . . . . 9 TopOn | |
7 | cnmpt21.a | . . . . . . . . 9 | |
8 | cnf2 12752 | . . . . . . . . 9 TopOn TopOn | |
9 | 5, 6, 7, 8 | syl3anc 1227 | . . . . . . . 8 |
10 | eqid 2164 | . . . . . . . . 9 | |
11 | 10 | fmpo 6161 | . . . . . . . 8 |
12 | 9, 11 | sylibr 133 | . . . . . . 7 |
13 | rsp2 2514 | . . . . . . 7 | |
14 | 12, 13 | syl 14 | . . . . . 6 |
15 | 14 | 3impib 1190 | . . . . 5 |
16 | cnmpt22.m | . . . . . . . . 9 TopOn | |
17 | cnmpt2t.b | . . . . . . . . 9 | |
18 | cnf2 12752 | . . . . . . . . 9 TopOn TopOn | |
19 | 5, 16, 17, 18 | syl3anc 1227 | . . . . . . . 8 |
20 | eqid 2164 | . . . . . . . . 9 | |
21 | 20 | fmpo 6161 | . . . . . . . 8 |
22 | 19, 21 | sylibr 133 | . . . . . . 7 |
23 | rsp2 2514 | . . . . . . 7 | |
24 | 22, 23 | syl 14 | . . . . . 6 |
25 | 24 | 3impib 1190 | . . . . 5 |
26 | 15, 25 | jca 304 | . . . . . 6 |
27 | txtopon 12809 | . . . . . . . . . . 11 TopOn TopOn TopOn | |
28 | 6, 16, 27 | syl2anc 409 | . . . . . . . . . 10 TopOn |
29 | cnmpt22.c | . . . . . . . . . . . 12 | |
30 | cntop2 12749 | . . . . . . . . . . . 12 | |
31 | 29, 30 | syl 14 | . . . . . . . . . . 11 |
32 | toptopon2 12564 | . . . . . . . . . . 11 TopOn | |
33 | 31, 32 | sylib 121 | . . . . . . . . . 10 TopOn |
34 | cnf2 12752 | . . . . . . . . . 10 TopOn TopOn | |
35 | 28, 33, 29, 34 | syl3anc 1227 | . . . . . . . . 9 |
36 | eqid 2164 | . . . . . . . . . 10 | |
37 | 36 | fmpo 6161 | . . . . . . . . 9 |
38 | 35, 37 | sylibr 133 | . . . . . . . 8 |
39 | r2al 2483 | . . . . . . . 8 | |
40 | 38, 39 | sylib 121 | . . . . . . 7 |
41 | 40 | 3ad2ant1 1007 | . . . . . 6 |
42 | eleq1 2227 | . . . . . . . . 9 | |
43 | eleq1 2227 | . . . . . . . . 9 | |
44 | 42, 43 | bi2anan9 596 | . . . . . . . 8 |
45 | cnmpt22.d | . . . . . . . . 9 | |
46 | 45 | eleq1d 2233 | . . . . . . . 8 |
47 | 44, 46 | imbi12d 233 | . . . . . . 7 |
48 | 47 | spc2gv 2812 | . . . . . 6 |
49 | 26, 41, 26, 48 | syl3c 63 | . . . . 5 |
50 | 45, 36 | ovmpoga 5962 | . . . . 5 |
51 | 15, 25, 49, 50 | syl3anc 1227 | . . . 4 |
52 | 1, 51 | eqtr3id 2211 | . . 3 |
53 | 52 | mpoeq3dva 5897 | . 2 |
54 | 2, 3, 7, 17 | cnmpt2t 12840 | . . 3 |
55 | 2, 3, 54, 29 | cnmpt21f 12839 | . 2 |
56 | 53, 55 | eqeltrrd 2242 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wal 1340 wceq 1342 wcel 2135 wral 2442 cop 3573 cuni 3783 cxp 4596 wf 5178 cfv 5182 (class class class)co 5836 cmpo 5838 ctop 12542 TopOnctopon 12555 ccn 12732 ctx 12799 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-topgen 12519 df-top 12543 df-topon 12556 df-bases 12588 df-cn 12735 df-tx 12800 |
This theorem is referenced by: cnmpt22f 12842 |
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