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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmapclsntr | Structured version Visualization version GIF version |
Description: The closure and interior operators on a topology are duals of each other. See also kur14lem2 34870. (Contributed by RP, 22-Apr-2021.) |
Ref | Expression |
---|---|
dssmapclsntr.x | β’ π = βͺ π½ |
dssmapclsntr.k | β’ πΎ = (clsβπ½) |
dssmapclsntr.i | β’ πΌ = (intβπ½) |
dssmapclsntr.o | β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π )))))) |
dssmapclsntr.d | β’ π· = (πβπ) |
Ref | Expression |
---|---|
dssmapclsntr | β’ (π½ β Top β πΎ = (π·βπΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dssmapclsntr.x | . . . . 5 β’ π = βͺ π½ | |
2 | dssmapclsntr.k | . . . . 5 β’ πΎ = (clsβπ½) | |
3 | dssmapclsntr.i | . . . . 5 β’ πΌ = (intβπ½) | |
4 | dssmapclsntr.o | . . . . 5 β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π )))))) | |
5 | dssmapclsntr.d | . . . . 5 β’ π· = (πβπ) | |
6 | 1, 2, 3, 4, 5 | dssmapntrcls 43619 | . . . 4 β’ (π½ β Top β πΌ = (π·βπΎ)) |
7 | 6 | eqcomd 2731 | . . 3 β’ (π½ β Top β (π·βπΎ) = πΌ) |
8 | 1 | topopn 22821 | . . . . 5 β’ (π½ β Top β π β π½) |
9 | 4, 5, 8 | dssmapf1od 43512 | . . . 4 β’ (π½ β Top β π·:(π« π βm π« π)β1-1-ontoβ(π« π βm π« π)) |
10 | 1, 2 | clselmap 43618 | . . . 4 β’ (π½ β Top β πΎ β (π« π βm π« π)) |
11 | f1ocnvfv 7281 | . . . 4 β’ ((π·:(π« π βm π« π)β1-1-ontoβ(π« π βm π« π) β§ πΎ β (π« π βm π« π)) β ((π·βπΎ) = πΌ β (β‘π·βπΌ) = πΎ)) | |
12 | 9, 10, 11 | syl2anc 582 | . . 3 β’ (π½ β Top β ((π·βπΎ) = πΌ β (β‘π·βπΌ) = πΎ)) |
13 | 7, 12 | mpd 15 | . 2 β’ (π½ β Top β (β‘π·βπΌ) = πΎ) |
14 | 4, 5, 8 | dssmapnvod 43511 | . . 3 β’ (π½ β Top β β‘π· = π·) |
15 | 14 | fveq1d 6892 | . 2 β’ (π½ β Top β (β‘π·βπΌ) = (π·βπΌ)) |
16 | 13, 15 | eqtr3d 2767 | 1 β’ (π½ β Top β πΎ = (π·βπΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3463 β cdif 3938 π« cpw 4599 βͺ cuni 4904 β¦ cmpt 5227 β‘ccnv 5672 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7413 βm cmap 8838 Topctop 22808 intcnt 22934 clsccl 22935 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-map 8840 df-top 22809 df-cld 22936 df-ntr 22937 df-cls 22938 |
This theorem is referenced by: (None) |
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