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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrf | Structured version Visualization version GIF version |
Description: The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.) |
Ref | Expression |
---|---|
ntrrn.x | β’ π = βͺ π½ |
ntrrn.i | β’ πΌ = (intβπ½) |
Ref | Expression |
---|---|
ntrf | β’ (π½ β Top β πΌ:π« πβΆπ½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vpwex 5336 | . . . . . 6 β’ π« π β V | |
2 | 1 | inex2 5279 | . . . . 5 β’ (π½ β© π« π ) β V |
3 | 2 | uniex 7682 | . . . 4 β’ βͺ (π½ β© π« π ) β V |
4 | eqid 2733 | . . . 4 β’ (π β π« π β¦ βͺ (π½ β© π« π )) = (π β π« π β¦ βͺ (π½ β© π« π )) | |
5 | 3, 4 | fnmpti 6648 | . . 3 β’ (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π |
6 | ntrrn.i | . . . . 5 β’ πΌ = (intβπ½) | |
7 | ntrrn.x | . . . . . 6 β’ π = βͺ π½ | |
8 | 7 | ntrfval 22398 | . . . . 5 β’ (π½ β Top β (intβπ½) = (π β π« π β¦ βͺ (π½ β© π« π ))) |
9 | 6, 8 | eqtrid 2785 | . . . 4 β’ (π½ β Top β πΌ = (π β π« π β¦ βͺ (π½ β© π« π ))) |
10 | 9 | fneq1d 6599 | . . 3 β’ (π½ β Top β (πΌ Fn π« π β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π)) |
11 | 5, 10 | mpbiri 258 | . 2 β’ (π½ β Top β πΌ Fn π« π) |
12 | 7, 6 | ntrrn 42486 | . 2 β’ (π½ β Top β ran πΌ β π½) |
13 | df-f 6504 | . 2 β’ (πΌ:π« πβΆπ½ β (πΌ Fn π« π β§ ran πΌ β π½)) | |
14 | 11, 12, 13 | sylanbrc 584 | 1 β’ (π½ β Top β πΌ:π« πβΆπ½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β© cin 3913 β wss 3914 π« cpw 4564 βͺ cuni 4869 β¦ cmpt 5192 ran crn 5638 Fn wfn 6495 βΆwf 6496 βcfv 6500 Topctop 22265 intcnt 22391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-top 22266 df-ntr 22394 |
This theorem is referenced by: ntrf2 42488 |
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