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Mirrors > Home > MPE Home > Th. List > cnmptid | Structured version Visualization version GIF version |
Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
cnmptid.j | β’ (π β π½ β (TopOnβπ)) |
Ref | Expression |
---|---|
cnmptid | β’ (π β (π₯ β π β¦ π₯) β (π½ Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptresid 6049 | . 2 β’ ( I βΎ π) = (π₯ β π β¦ π₯) | |
2 | cnmptid.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
3 | idcn 22981 | . . 3 β’ (π½ β (TopOnβπ) β ( I βΎ π) β (π½ Cn π½)) | |
4 | 2, 3 | syl 17 | . 2 β’ (π β ( I βΎ π) β (π½ Cn π½)) |
5 | 1, 4 | eqeltrrid 2836 | 1 β’ (π β (π₯ β π β¦ π₯) β (π½ Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2104 β¦ cmpt 5230 I cid 5572 βΎ cres 5677 βcfv 6542 (class class class)co 7411 TopOnctopon 22632 Cn ccn 22948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-map 8824 df-top 22616 df-topon 22633 df-cn 22951 |
This theorem is referenced by: xkoinjcn 23411 txconn 23413 imasnopn 23414 imasncld 23415 imasncls 23416 pt1hmeo 23530 istgp2 23815 tmdmulg 23816 tmdlactcn 23826 clsnsg 23834 tgpt0 23843 tlmtgp 23920 nmcn 24580 expcn 24610 divccn 24611 expcnOLD 24612 divccnOLD 24613 cncfmptid 24653 cdivcncf 24661 iirevcn 24671 iihalf1cn 24673 iihalf1cnOLD 24674 iihalf2cn 24676 iihalf2cnOLD 24677 icchmeo 24685 icchmeoOLD 24686 evth2 24706 pcocn 24764 pcopt 24769 pcopt2 24770 pcoass 24771 csscld 24997 clsocv 24998 dvcnvlem 25728 resqrtcn 26493 sqrtcn 26494 efrlim 26710 ipasslem7 30356 occllem 30823 hmopidmchi 31671 rmulccn 33206 cxpcncf1 33905 cvxpconn 34531 cvmlift2lem2 34593 cvmlift2lem3 34594 cvmliftphtlem 34606 gg-rmulccn 35465 knoppcnlem10 35681 cxpcncf2 44913 |
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