![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > neircl | Structured version Visualization version GIF version |
Description: Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.) |
Ref | Expression |
---|---|
neircl | β’ (π β ((neiβπ½)βπ) β π½ β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvne0 47771 | . 2 β’ (π β ((neiβπ½)βπ) β (neiβπ½) β β ) | |
2 | n0 4341 | . . 3 β’ ((neiβπ½) β β β βπ π β (neiβπ½)) | |
3 | 2 | biimpi 215 | . 2 β’ ((neiβπ½) β β β βπ π β (neiβπ½)) |
4 | df-nei 22952 | . . . 4 β’ nei = (π β Top β¦ (π₯ β π« βͺ π β¦ {π¦ β π« βͺ π β£ βπ β π (π₯ β π β§ π β π¦)})) | |
5 | 4 | mptrcl 7000 | . . 3 β’ (π β (neiβπ½) β π½ β Top) |
6 | 5 | exlimiv 1925 | . 2 β’ (βπ π β (neiβπ½) β π½ β Top) |
7 | 1, 3, 6 | 3syl 18 | 1 β’ (π β ((neiβπ½)βπ) β π½ β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 βwex 1773 β wcel 2098 β wne 2934 βwrex 3064 {crab 3426 β wss 3943 β c0 4317 π« cpw 4597 βͺ cuni 4902 β¦ cmpt 5224 βcfv 6536 Topctop 22745 neicnei 22951 |
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-sep 5292 ax-nul 5299 ax-pr 5420 |
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-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fv 6544 df-nei 22952 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |