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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 3922 | . 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 42471 | . . . . . 6 β’ (π β π β (π« π« π΅ βm π΅)) |
9 | elmapi 8793 | . . . . . 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 7040 | . . . 4 β’ (π β (πβπ) β π« π« π΅) |
13 | 12 | elpwid 4573 | . . 3 β’ (π β (πβπ) β π« π΅) |
14 | sseqin2 4179 | . . 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 482 | . . . . 5 β’ ((π β§ π β π« π΅) β πΎπ»π) |
17 | 11 | adantr 482 | . . . . 5 β’ ((π β§ π β π« π΅) β π β π΅) |
18 | simpr 486 | . . . . 5 β’ ((π β§ π β π« π΅) β π β π« π΅) | |
19 | 2, 3, 4, 5, 6, 16, 17, 18 | clsneiel2 42473 | . . . 4 β’ ((π β§ π β π« π΅) β (π β (πΎβ(π΅ β π )) β Β¬ π β (πβπ))) |
20 | 19 | con2bid 355 | . . 3 β’ ((π β§ π β π« π΅) β (π β (πβπ) β Β¬ π β (πΎβ(π΅ β π )))) |
21 | 20 | rabbidva 3413 | . 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 2797 | 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 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3447 β cdif 3911 β© cin 3913 β wss 3914 π« cpw 4564 class class class wbr 5109 β¦ cmpt 5192 β ccom 5641 βΆwf 6496 βcfv 6500 (class class class)co 7361 β cmpo 7363 βm cmap 8771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 |
This theorem is referenced by: (None) |
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