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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 47979 | . 2 β’ (π β ((neiβπ½)βπ) β (neiβπ½) β β ) | |
2 | n0 4350 | . . 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 23022 | . . . 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 7019 | . . 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 394 βwex 1773 β wcel 2098 β wne 2937 βwrex 3067 {crab 3430 β wss 3949 β c0 4326 π« cpw 4606 βͺ cuni 4912 β¦ cmpt 5235 βcfv 6553 Topctop 22815 neicnei 23021 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fv 6561 df-nei 23022 |
This theorem is referenced by: (None) |
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