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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 12996 | . . . 4 | |
6 | 3, 5 | syl 14 | . . 3 |
7 | toptopon2 12811 | . . 3 TopOn | |
8 | 6, 7 | sylib 121 | . 2 TopOn |
9 | cntop2 12996 | . . . 4 | |
10 | 4, 9 | syl 14 | . . 3 |
11 | toptopon2 12811 | . . 3 TopOn | |
12 | 10, 11 | sylib 121 | . 2 TopOn |
13 | txtopon 13056 | . . . . . . 7 TopOn TopOn TopOn | |
14 | 8, 12, 13 | syl2anc 409 | . . . . . 6 TopOn |
15 | cnmpt22f.f | . . . . . . . 8 | |
16 | cntop2 12996 | . . . . . . . 8 | |
17 | 15, 16 | syl 14 | . . . . . . 7 |
18 | toptopon2 12811 | . . . . . . 7 TopOn | |
19 | 17, 18 | sylib 121 | . . . . . 6 TopOn |
20 | cnf2 12999 | . . . . . 6 TopOn TopOn | |
21 | 14, 19, 15, 20 | syl3anc 1233 | . . . . 5 |
22 | 21 | ffnd 5348 | . . . 4 |
23 | fnovim 5961 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | 24, 15 | eqeltrrd 2248 | . 2 |
26 | oveq12 5862 | . 2 | |
27 | 1, 2, 3, 4, 8, 12, 25, 26 | cnmpt22 13088 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 cuni 3796 cxp 4609 wfn 5193 wf 5194 cfv 5198 (class class class)co 5853 cmpo 5855 ctop 12789 TopOnctopon 12802 ccn 12979 ctx 13046 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-topgen 12600 df-top 12790 df-topon 12803 df-bases 12835 df-cn 12982 df-tx 13047 |
This theorem is referenced by: cnmptcom 13092 divcnap 13349 cnrehmeocntop 13387 |
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