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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneifv3 | Structured version Visualization version GIF version |
Description: Value of the neighborhoods (convergents) in terms of the closure (interior) 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.x | β’ (π β π β π΅) |
Ref | Expression |
---|---|
clsneifv3 | β’ (π β (πβπ) = {π β π« π΅ β£ Β¬ π β (πΎβ(π΅ β π ))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3955 | . 2 β’ (π« π΅ β© (πβπ)) = {π β π« π΅ β£ π β (πβπ)} | |
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 | clsneinex 43537 | . . . . . 6 β’ (π β π β (π« π« π΅ βm π΅)) |
9 | elmapi 8868 | . . . . . 6 β’ (π β (π« π« π΅ βm π΅) β π:π΅βΆπ« π« π΅) | |
10 | 8, 9 | syl 17 | . . . . 5 β’ (π β π:π΅βΆπ« π« π΅) |
11 | clsneifv.x | . . . . 5 β’ (π β π β π΅) | |
12 | 10, 11 | ffvelcdmd 7095 | . . . 4 β’ (π β (πβπ) β π« π« π΅) |
13 | 12 | elpwid 4612 | . . 3 β’ (π β (πβπ) β π« π΅) |
14 | sseqin2 4215 | . . 3 β’ ((πβπ) β π« π΅ β (π« π΅ β© (πβπ)) = (πβπ)) | |
15 | 13, 14 | sylib 217 | . 2 β’ (π β (π« π΅ β© (πβπ)) = (πβπ)) |
16 | 7 | adantr 480 | . . . . 5 β’ ((π β§ π β π« π΅) β πΎπ»π) |
17 | 11 | adantr 480 | . . . . 5 β’ ((π β§ π β π« π΅) β π β π΅) |
18 | simpr 484 | . . . . 5 β’ ((π β§ π β π« π΅) β π β π« π΅) | |
19 | 2, 3, 4, 5, 6, 16, 17, 18 | clsneiel2 43539 | . . . 4 β’ ((π β§ π β π« π΅) β (π β (πΎβ(π΅ β π )) β Β¬ π β (πβπ))) |
20 | 19 | con2bid 354 | . . 3 β’ ((π β§ π β π« π΅) β (π β (πβπ) β Β¬ π β (πΎβ(π΅ β π )))) |
21 | 20 | rabbidva 3436 | . 2 β’ (π β {π β π« π΅ β£ π β (πβπ)} = {π β π« π΅ β£ Β¬ π β (πΎβ(π΅ β π ))}) |
22 | 1, 15, 21 | 3eqtr3a 2792 | 1 β’ (π β (πβπ) = {π β π« π΅ β£ Β¬ π β (πΎβ(π΅ β π ))}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 {crab 3429 Vcvv 3471 β cdif 3944 β© cin 3946 β wss 3947 π« cpw 4603 class class class wbr 5148 β¦ cmpt 5231 β ccom 5682 βΆwf 6544 βcfv 6548 (class class class)co 7420 β cmpo 7422 βm cmap 8845 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8847 |
This theorem is referenced by: (None) |
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