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Mirrors > Home > MPE Home > Th. List > cnmpt1res | Structured version Visualization version GIF version |
Description: The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
cnmpt1res.2 | β’ πΎ = (π½ βΎt π) |
cnmpt1res.3 | β’ (π β π½ β (TopOnβπ)) |
cnmpt1res.5 | β’ (π β π β π) |
cnmpt1res.6 | β’ (π β (π₯ β π β¦ π΄) β (π½ Cn πΏ)) |
Ref | Expression |
---|---|
cnmpt1res | β’ (π β (π₯ β π β¦ π΄) β (πΎ Cn πΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmpt1res.5 | . . 3 β’ (π β π β π) | |
2 | 1 | resmptd 5998 | . 2 β’ (π β ((π₯ β π β¦ π΄) βΎ π) = (π₯ β π β¦ π΄)) |
3 | cnmpt1res.6 | . . . 4 β’ (π β (π₯ β π β¦ π΄) β (π½ Cn πΏ)) | |
4 | cnmpt1res.3 | . . . . . 6 β’ (π β π½ β (TopOnβπ)) | |
5 | toponuni 22286 | . . . . . 6 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
6 | 4, 5 | syl 17 | . . . . 5 β’ (π β π = βͺ π½) |
7 | 1, 6 | sseqtrd 3988 | . . . 4 β’ (π β π β βͺ π½) |
8 | eqid 2733 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
9 | 8 | cnrest 22659 | . . . 4 β’ (((π₯ β π β¦ π΄) β (π½ Cn πΏ) β§ π β βͺ π½) β ((π₯ β π β¦ π΄) βΎ π) β ((π½ βΎt π) Cn πΏ)) |
10 | 3, 7, 9 | syl2anc 585 | . . 3 β’ (π β ((π₯ β π β¦ π΄) βΎ π) β ((π½ βΎt π) Cn πΏ)) |
11 | cnmpt1res.2 | . . . 4 β’ πΎ = (π½ βΎt π) | |
12 | 11 | oveq1i 7371 | . . 3 β’ (πΎ Cn πΏ) = ((π½ βΎt π) Cn πΏ) |
13 | 10, 12 | eleqtrrdi 2845 | . 2 β’ (π β ((π₯ β π β¦ π΄) βΎ π) β (πΎ Cn πΏ)) |
14 | 2, 13 | eqeltrrd 2835 | 1 β’ (π β (π₯ β π β¦ π΄) β (πΎ Cn πΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3914 βͺ cuni 4869 β¦ cmpt 5192 βΎ cres 5639 βcfv 6500 (class class class)co 7361 βΎt crest 17310 TopOnctopon 22282 Cn ccn 22598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-map 8773 df-en 8890 df-fin 8893 df-fi 9355 df-rest 17312 df-topgen 17333 df-top 22266 df-topon 22283 df-bases 22319 df-cn 22601 |
This theorem is referenced by: subgtgp 23479 symgtgp 23480 cnmptre 24313 evth2 24346 pcoass 24410 efrlim 26342 ipasslem7 29827 cvxpconn 33900 cvmliftlem8 33950 |
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