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Mirrors > Home > ILE Home > Th. List > ntrfval | GIF version |
Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
cldval.1 | β’ π = βͺ π½ |
Ref | Expression |
---|---|
ntrfval | β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldval.1 | . . . 4 β’ π = βͺ π½ | |
2 | 1 | topopn 13511 | . . 3 β’ (π½ β Top β π β π½) |
3 | pwexg 4181 | . . 3 β’ (π β π½ β π« π β V) | |
4 | mptexg 5742 | . . 3 β’ (π« π β V β (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) β V) | |
5 | 2, 3, 4 | 3syl 17 | . 2 β’ (π½ β Top β (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) β V) |
6 | unieq 3819 | . . . . . 6 β’ (π = π½ β βͺ π = βͺ π½) | |
7 | 6, 1 | eqtr4di 2228 | . . . . 5 β’ (π = π½ β βͺ π = π) |
8 | 7 | pweqd 3581 | . . . 4 β’ (π = π½ β π« βͺ π = π« π) |
9 | ineq1 3330 | . . . . 5 β’ (π = π½ β (π β© π« π₯) = (π½ β© π« π₯)) | |
10 | 9 | unieqd 3821 | . . . 4 β’ (π = π½ β βͺ (π β© π« π₯) = βͺ (π½ β© π« π₯)) |
11 | 8, 10 | mpteq12dv 4086 | . . 3 β’ (π = π½ β (π₯ β π« βͺ π β¦ βͺ (π β© π« π₯)) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
12 | df-ntr 13599 | . . 3 β’ int = (π β Top β¦ (π₯ β π« βͺ π β¦ βͺ (π β© π« π₯))) | |
13 | 11, 12 | fvmptg 5593 | . 2 β’ ((π½ β Top β§ (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) β V) β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
14 | 5, 13 | mpdan 421 | 1 β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 Vcvv 2738 β© cin 3129 π« cpw 3576 βͺ cuni 3810 β¦ cmpt 4065 βcfv 5217 Topctop 13500 intcnt 13596 |
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 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
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 2740 df-sbc 2964 df-csb 3059 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-top 13501 df-ntr 13599 |
This theorem is referenced by: ntrval 13613 |
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