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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 22801 | . . . . 5 β’ (π½ β Top β π β π½) |
3 | pwexg 5372 | . . . . 5 β’ (π β π½ β π« π β V) | |
4 | rabexg 5327 | . . . . 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 3146 | . . 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 2728 | . . . 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 6689 | . . 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 22996 | . . 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 6641 | . 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 1534 β wcel 2099 βwral 3057 βwrex 3066 {crab 3428 Vcvv 3470 β wss 3945 π« cpw 4598 βͺ cuni 4903 β¦ cmpt 5225 Fn wfn 6537 βcfv 6542 Topctop 22788 neicnei 22994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22789 df-nei 22995 |
This theorem is referenced by: neiss2 22998 |
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