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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 6044 | . 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 22829 | . . . . . 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 4020 | . . . 4 β’ (π β π β βͺ π½) |
8 | eqid 2728 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
9 | 8 | cnrest 23202 | . . . 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 583 | . . 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 7430 | . . 3 β’ (πΎ Cn πΏ) = ((π½ βΎt π) Cn πΏ) |
13 | 10, 12 | eleqtrrdi 2840 | . 2 β’ (π β ((π₯ β π β¦ π΄) βΎ π) β (πΎ Cn πΏ)) |
14 | 2, 13 | eqeltrrd 2830 | 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 1534 β wcel 2099 β wss 3947 βͺ cuni 4908 β¦ cmpt 5231 βΎ cres 5680 βcfv 6548 (class class class)co 7420 βΎt crest 17402 TopOnctopon 22825 Cn ccn 23141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-map 8847 df-en 8965 df-fin 8968 df-fi 9435 df-rest 17404 df-topgen 17425 df-top 22809 df-topon 22826 df-bases 22862 df-cn 23144 |
This theorem is referenced by: subgtgp 24022 symgtgp 24023 cnmptre 24861 evth2 24899 pcoass 24964 efrlim 26914 efrlimOLD 26915 ipasslem7 30659 cvxpconn 34852 cvmliftlem8 34902 |
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