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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 34740. (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 43471 | . . . 4 β’ (π½ β Top β πΌ = (π·βπΎ)) |
7 | 6 | eqcomd 2733 | . . 3 β’ (π½ β Top β (π·βπΎ) = πΌ) |
8 | 1 | topopn 22782 | . . . . 5 β’ (π½ β Top β π β π½) |
9 | 4, 5, 8 | dssmapf1od 43364 | . . . 4 β’ (π½ β Top β π·:(π« π βm π« π)β1-1-ontoβ(π« π βm π« π)) |
10 | 1, 2 | clselmap 43470 | . . . 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 583 | . . 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 43363 | . . 3 β’ (π½ β Top β β‘π· = π·) |
15 | 14 | fveq1d 6893 | . 2 β’ (π½ β Top β (β‘π·βπΌ) = (π·βπΌ)) |
16 | 13, 15 | eqtr3d 2769 | 1 β’ (π½ β Top β πΎ = (π·βπΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3469 β cdif 3941 π« cpw 4598 βͺ cuni 4903 β¦ cmpt 5225 β‘ccnv 5671 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 βm cmap 8834 Topctop 22769 intcnt 22895 clsccl 22896 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-map 8836 df-top 22770 df-cld 22897 df-ntr 22898 df-cls 22899 |
This theorem is referenced by: (None) |
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