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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 22828 | . . 3 β’ (π½ β Top β π β π½) |
3 | pwexg 5382 | . . 3 β’ (π β π½ β π« π β V) | |
4 | mptexg 7239 | . . 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 4923 | . . . . . 6 β’ (π = π½ β βͺ π = βͺ π½) | |
7 | 6, 1 | eqtr4di 2786 | . . . . 5 β’ (π = π½ β βͺ π = π) |
8 | 7 | pweqd 4623 | . . . 4 β’ (π = π½ β π« βͺ π = π« π) |
9 | ineq1 4207 | . . . . 5 β’ (π = π½ β (π β© π« π₯) = (π½ β© π« π₯)) | |
10 | 9 | unieqd 4925 | . . . 4 β’ (π = π½ β βͺ (π β© π« π₯) = βͺ (π½ β© π« π₯)) |
11 | 8, 10 | mpteq12dv 5243 | . . 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 22944 | . . 3 β’ int = (π β Top β¦ (π₯ β π« βͺ π β¦ βͺ (π β© π« π₯))) | |
13 | 11, 12 | fvmptg 7008 | . 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 685 | 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 3473 β© cin 3948 π« cpw 4606 βͺ cuni 4912 β¦ cmpt 5235 βcfv 6553 Topctop 22815 intcnt 22941 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-top 22816 df-ntr 22944 |
This theorem is referenced by: ntrval 22960 ntrrn 43583 ntrf 43584 dssmapntrcls 43589 |
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