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Mirrors > Home > ILE Home > Th. List > cnmpt1t | 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 |
---|---|
cnmptid.j | TopOn |
cnmpt11.a | |
cnmpt1t.b |
Ref | Expression |
---|---|
cnmpt1t |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmptid.j | . . . 4 TopOn | |
2 | toponuni 12182 | . . . 4 TopOn | |
3 | mpteq1 4012 | . . . 4 | |
4 | 1, 2, 3 | 3syl 17 | . . 3 |
5 | simpr 109 | . . . . . 6 | |
6 | cnmpt11.a | . . . . . . . . . 10 | |
7 | cntop2 12371 | . . . . . . . . . 10 | |
8 | 6, 7 | syl 14 | . . . . . . . . 9 |
9 | toptopon2 12186 | . . . . . . . . 9 TopOn | |
10 | 8, 9 | sylib 121 | . . . . . . . 8 TopOn |
11 | cnf2 12374 | . . . . . . . 8 TopOn TopOn | |
12 | 1, 10, 6, 11 | syl3anc 1216 | . . . . . . 7 |
13 | 12 | fvmptelrn 5573 | . . . . . 6 |
14 | eqid 2139 | . . . . . . 7 | |
15 | 14 | fvmpt2 5504 | . . . . . 6 |
16 | 5, 13, 15 | syl2anc 408 | . . . . 5 |
17 | cnmpt1t.b | . . . . . . . . . 10 | |
18 | cntop2 12371 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | toptopon2 12186 | . . . . . . . . 9 TopOn | |
21 | 19, 20 | sylib 121 | . . . . . . . 8 TopOn |
22 | cnf2 12374 | . . . . . . . 8 TopOn TopOn | |
23 | 1, 21, 17, 22 | syl3anc 1216 | . . . . . . 7 |
24 | 23 | fvmptelrn 5573 | . . . . . 6 |
25 | eqid 2139 | . . . . . . 7 | |
26 | 25 | fvmpt2 5504 | . . . . . 6 |
27 | 5, 24, 26 | syl2anc 408 | . . . . 5 |
28 | 16, 27 | opeq12d 3713 | . . . 4 |
29 | 28 | mpteq2dva 4018 | . . 3 |
30 | 4, 29 | eqtr3d 2174 | . 2 |
31 | eqid 2139 | . . . 4 | |
32 | nfcv 2281 | . . . . 5 | |
33 | nffvmpt1 5432 | . . . . . 6 | |
34 | nffvmpt1 5432 | . . . . . 6 | |
35 | 33, 34 | nfop 3721 | . . . . 5 |
36 | fveq2 5421 | . . . . . 6 | |
37 | fveq2 5421 | . . . . . 6 | |
38 | 36, 37 | opeq12d 3713 | . . . . 5 |
39 | 32, 35, 38 | cbvmpt 4023 | . . . 4 |
40 | 31, 39 | txcnmpt 12442 | . . 3 |
41 | 6, 17, 40 | syl2anc 408 | . 2 |
42 | 30, 41 | eqeltrrd 2217 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cop 3530 cuni 3736 cmpt 3989 wf 5119 cfv 5123 (class class class)co 5774 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: cnmpt12f 12455 imasnopn 12468 cnrehmeocntop 12762 |
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