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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 6033 | . 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 22767 | . . . . . 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 4017 | . . . 4 β’ (π β π β βͺ π½) |
8 | eqid 2726 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
9 | 8 | cnrest 23140 | . . . 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 7414 | . . 3 β’ (πΎ Cn πΏ) = ((π½ βΎt π) Cn πΏ) |
13 | 10, 12 | eleqtrrdi 2838 | . 2 β’ (π β ((π₯ β π β¦ π΄) βΎ π) β (πΎ Cn πΏ)) |
14 | 2, 13 | eqeltrrd 2828 | 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 1533 β wcel 2098 β wss 3943 βͺ cuni 4902 β¦ cmpt 5224 βΎ cres 5671 βcfv 6536 (class class class)co 7404 βΎt crest 17373 TopOnctopon 22763 Cn ccn 23079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-map 8821 df-en 8939 df-fin 8942 df-fi 9405 df-rest 17375 df-topgen 17396 df-top 22747 df-topon 22764 df-bases 22800 df-cn 23082 |
This theorem is referenced by: subgtgp 23960 symgtgp 23961 cnmptre 24799 evth2 24837 pcoass 24902 efrlim 26852 efrlimOLD 26853 ipasslem7 30594 cvxpconn 34761 cvmliftlem8 34811 |
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