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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 47677 | . 2 β’ (π β ((neiβπ½)βπ) β (neiβπ½) β β ) | |
2 | n0 4346 | . . 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 22922 | . . . 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 7007 | . . 3 β’ (π β (neiβπ½) β π½ β Top) |
6 | 5 | exlimiv 1932 | . 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 1780 β wcel 2105 β wne 2939 βwrex 3069 {crab 3431 β wss 3948 β c0 4322 π« cpw 4602 βͺ cuni 4908 β¦ cmpt 5231 βcfv 6543 Topctop 22715 neicnei 22921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fv 6551 df-nei 22922 |
This theorem is referenced by: (None) |
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