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Mirrors > Home > MPE Home > Th. List > ntrfval | Structured version Visualization version 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 22758 | . . 3 β’ (π½ β Top β π β π½) |
3 | pwexg 5369 | . . 3 β’ (π β π½ β π« π β V) | |
4 | mptexg 7217 | . . 3 β’ (π« π β V β (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) β V) | |
5 | 2, 3, 4 | 3syl 18 | . 2 β’ (π½ β Top β (π₯ β π« π β¦ βͺ (π½ β© π« π₯)) β V) |
6 | unieq 4913 | . . . . . 6 β’ (π = π½ β βͺ π = βͺ π½) | |
7 | 6, 1 | eqtr4di 2784 | . . . . 5 β’ (π = π½ β βͺ π = π) |
8 | 7 | pweqd 4614 | . . . 4 β’ (π = π½ β π« βͺ π = π« π) |
9 | ineq1 4200 | . . . . 5 β’ (π = π½ β (π β© π« π₯) = (π½ β© π« π₯)) | |
10 | 9 | unieqd 4915 | . . . 4 β’ (π = π½ β βͺ (π β© π« π₯) = βͺ (π½ β© π« π₯)) |
11 | 8, 10 | mpteq12dv 5232 | . . 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 22874 | . . 3 β’ int = (π β Top β¦ (π₯ β π« βͺ π β¦ βͺ (π β© π« π₯))) | |
13 | 11, 12 | fvmptg 6989 | . 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 684 | 1 β’ (π½ β Top β (intβπ½) = (π₯ β π« π β¦ βͺ (π½ β© π« π₯))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3468 β© cin 3942 π« cpw 4597 βͺ cuni 4902 β¦ cmpt 5224 βcfv 6536 Topctop 22745 intcnt 22871 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-top 22746 df-ntr 22874 |
This theorem is referenced by: ntrval 22890 ntrrn 43431 ntrf 43432 dssmapntrcls 43437 |
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