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Mirrors > Home > ILE Home > Th. List > neif | 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 13593 | . . . . 5 β’ (π½ β Top β π β π½) |
3 | pwexg 4182 | . . . . 5 β’ (π β π½ β π« π β V) | |
4 | rabexg 4148 | . . . . 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 17 | . . . 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 2551 | . . 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 2177 | . . . 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 5344 | . . 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 14 | . 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 13725 | . . 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 5308 | . 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 167 | 1 β’ (π½ β Top β (neiβπ½) Fn π« π) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βwral 2455 βwrex 2456 {crab 2459 Vcvv 2739 β wss 3131 π« cpw 3577 βͺ cuni 3811 β¦ cmpt 4066 Fn wfn 5213 βcfv 5218 Topctop 13582 neicnei 13723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-top 13583 df-nei 13724 |
This theorem is referenced by: neiss2 13727 |
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