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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneifv4 | Structured version Visualization version GIF version |
Description: Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.) |
Ref | Expression |
---|---|
clsnei.o | β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) |
clsnei.p | 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)))))) |
clsnei.d | β’ π· = (πβπ΅) |
clsnei.f | β’ πΉ = (π« π΅ππ΅) |
clsnei.h | β’ π» = (πΉ β π·) |
clsnei.r | β’ (π β πΎπ»π) |
clsneifv.s | β’ (π β π β π« π΅) |
Ref | Expression |
---|---|
clsneifv4 | β’ (π β (πΎβπ) = {π₯ β π΅ β£ Β¬ (π΅ β π) β (πβπ₯)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3951 | . 2 β’ (π΅ β© (πΎβπ)) = {π₯ β π΅ β£ π₯ β (πΎβπ)} | |
2 | clsnei.o | . . . . . . 7 β’ π = (π β V, π β V β¦ (π β (π« π βm π) β¦ (π β π β¦ {π β π β£ π β (πβπ)}))) | |
3 | clsnei.p | . . . . . . 7 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)))))) | |
4 | clsnei.d | . . . . . . 7 β’ π· = (πβπ΅) | |
5 | clsnei.f | . . . . . . 7 β’ πΉ = (π« π΅ππ΅) | |
6 | clsnei.h | . . . . . . 7 β’ π» = (πΉ β π·) | |
7 | clsnei.r | . . . . . . 7 β’ (π β πΎπ»π) | |
8 | 2, 3, 4, 5, 6, 7 | clsneikex 43414 | . . . . . 6 β’ (π β πΎ β (π« π΅ βm π« π΅)) |
9 | elmapi 8842 | . . . . . 6 β’ (πΎ β (π« π΅ βm π« π΅) β πΎ:π« π΅βΆπ« π΅) | |
10 | 8, 9 | syl 17 | . . . . 5 β’ (π β πΎ:π« π΅βΆπ« π΅) |
11 | clsneifv.s | . . . . 5 β’ (π β π β π« π΅) | |
12 | 10, 11 | ffvelcdmd 7080 | . . . 4 β’ (π β (πΎβπ) β π« π΅) |
13 | 12 | elpwid 4606 | . . 3 β’ (π β (πΎβπ) β π΅) |
14 | sseqin2 4210 | . . 3 β’ ((πΎβπ) β π΅ β (π΅ β© (πΎβπ)) = (πΎβπ)) | |
15 | 13, 14 | sylib 217 | . 2 β’ (π β (π΅ β© (πΎβπ)) = (πΎβπ)) |
16 | 7 | adantr 480 | . . . 4 β’ ((π β§ π₯ β π΅) β πΎπ»π) |
17 | simpr 484 | . . . 4 β’ ((π β§ π₯ β π΅) β π₯ β π΅) | |
18 | 11 | adantr 480 | . . . 4 β’ ((π β§ π₯ β π΅) β π β π« π΅) |
19 | 2, 3, 4, 5, 6, 16, 17, 18 | clsneiel1 43416 | . . 3 β’ ((π β§ π₯ β π΅) β (π₯ β (πΎβπ) β Β¬ (π΅ β π) β (πβπ₯))) |
20 | 19 | rabbidva 3433 | . 2 β’ (π β {π₯ β π΅ β£ π₯ β (πΎβπ)} = {π₯ β π΅ β£ Β¬ (π΅ β π) β (πβπ₯)}) |
21 | 1, 15, 20 | 3eqtr3a 2790 | 1 β’ (π β (πΎβπ) = {π₯ β π΅ β£ Β¬ (π΅ β π) β (πβπ₯)}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 Vcvv 3468 β cdif 3940 β© cin 3942 β wss 3943 π« cpw 4597 class class class wbr 5141 β¦ cmpt 5224 β ccom 5673 βΆwf 6532 βcfv 6536 (class class class)co 7404 β cmpo 7406 βm cmap 8819 |
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-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-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 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-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-1st 7971 df-2nd 7972 df-map 8821 |
This theorem is referenced by: (None) |
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