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Mirrors > Home > MPE Home > Th. List > neif | Structured version Visualization version GIF version |
Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
neifval.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
neif | β’ (π½ β Top β (neiβπ½) Fn π« π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 | . . . . . 6 β’ π = βͺ π½ | |
2 | 1 | topopn 22732 | . . . . 5 β’ (π½ β Top β π β π½) |
3 | pwexg 5367 | . . . . 5 β’ (π β π½ β π« π β V) | |
4 | rabexg 5322 | . . . . 5 β’ (π« π β V β {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)} β V) | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 β’ (π½ β Top β {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)} β V) |
6 | 5 | ralrimivw 3142 | . . 3 β’ (π½ β Top β βπ₯ β π« π{π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)} β V) |
7 | eqid 2724 | . . . 4 β’ (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)}) = (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)}) | |
8 | 7 | fnmpt 6681 | . . 3 β’ (βπ₯ β π« π{π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)} β V β (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)}) Fn π« π) |
9 | 6, 8 | syl 17 | . 2 β’ (π½ β Top β (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)}) Fn π« π) |
10 | 1 | neifval 22927 | . . 3 β’ (π½ β Top β (neiβπ½) = (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)})) |
11 | 10 | fneq1d 6633 | . 2 β’ (π½ β Top β ((neiβπ½) Fn π« π β (π₯ β π« π β¦ {π£ β π« π β£ βπ β π½ (π₯ β π β§ π β π£)}) Fn π« π)) |
12 | 9, 11 | mpbird 257 | 1 β’ (π½ β Top β (neiβπ½) Fn π« π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 βwrex 3062 {crab 3424 Vcvv 3466 β wss 3941 π« cpw 4595 βͺ cuni 4900 β¦ cmpt 5222 Fn wfn 6529 βcfv 6534 Topctop 22719 neicnei 22925 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-top 22720 df-nei 22926 |
This theorem is referenced by: neiss2 22929 |
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