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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 12644 | . . . 4 | |
6 | 3, 5 | syl 14 | . . 3 |
7 | toptopon2 12459 | . . 3 TopOn | |
8 | 6, 7 | sylib 121 | . 2 TopOn |
9 | cntop2 12644 | . . . 4 | |
10 | 4, 9 | syl 14 | . . 3 |
11 | toptopon2 12459 | . . 3 TopOn | |
12 | 10, 11 | sylib 121 | . 2 TopOn |
13 | txtopon 12704 | . . . . . . 7 TopOn TopOn TopOn | |
14 | 8, 12, 13 | syl2anc 409 | . . . . . 6 TopOn |
15 | cnmpt22f.f | . . . . . . . 8 | |
16 | cntop2 12644 | . . . . . . . 8 | |
17 | 15, 16 | syl 14 | . . . . . . 7 |
18 | toptopon2 12459 | . . . . . . 7 TopOn | |
19 | 17, 18 | sylib 121 | . . . . . 6 TopOn |
20 | cnf2 12647 | . . . . . 6 TopOn TopOn | |
21 | 14, 19, 15, 20 | syl3anc 1220 | . . . . 5 |
22 | 21 | ffnd 5321 | . . . 4 |
23 | fnovim 5930 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | 24, 15 | eqeltrrd 2235 | . 2 |
26 | oveq12 5834 | . 2 | |
27 | 1, 2, 3, 4, 8, 12, 25, 26 | cnmpt22 12736 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wcel 2128 cuni 3773 cxp 4585 wfn 5166 wf 5167 cfv 5171 (class class class)co 5825 cmpo 5827 ctop 12437 TopOnctopon 12450 ccn 12627 ctx 12694 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-map 6596 df-topgen 12414 df-top 12438 df-topon 12451 df-bases 12483 df-cn 12630 df-tx 12695 |
This theorem is referenced by: cnmptcom 12740 divcnap 12997 cnrehmeocntop 13035 |
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