![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cnmpt1res | 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 4960 | . 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 13600 | . . . . . 6 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
6 | 4, 5 | syl 14 | . . . . 5 β’ (π β π = βͺ π½) |
7 | 1, 6 | sseqtrd 3195 | . . . 4 β’ (π β π β βͺ π½) |
8 | eqid 2177 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
9 | 8 | cnrest 13820 | . . . 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 411 | . . 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 5887 | . . 3 β’ (πΎ Cn πΏ) = ((π½ βΎt π) Cn πΏ) |
13 | 10, 12 | eleqtrrdi 2271 | . 2 β’ (π β ((π₯ β π β¦ π΄) βΎ π) β (πΎ Cn πΏ)) |
14 | 2, 13 | eqeltrrd 2255 | 1 β’ (π β (π₯ β π β¦ π΄) β (πΎ Cn πΏ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 β wss 3131 βͺ cuni 3811 β¦ cmpt 4066 βΎ cres 4630 βcfv 5218 (class class class)co 5877 βΎt crest 12693 TopOnctopon 13595 Cn ccn 13770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-map 6652 df-rest 12695 df-topgen 12714 df-top 13583 df-topon 13596 df-bases 13628 df-cn 13773 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |