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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 22271 | . . 3 β’ (π½ β Top β π β π½) |
3 | pwexg 5334 | . . 3 β’ (π β π½ β π« π β V) | |
4 | mptexg 7172 | . . 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 4877 | . . . . . 6 β’ (π = π½ β βͺ π = βͺ π½) | |
7 | 6, 1 | eqtr4di 2791 | . . . . 5 β’ (π = π½ β βͺ π = π) |
8 | 7 | pweqd 4578 | . . . 4 β’ (π = π½ β π« βͺ π = π« π) |
9 | ineq1 4166 | . . . . 5 β’ (π = π½ β (π β© π« π₯) = (π½ β© π« π₯)) | |
10 | 9 | unieqd 4880 | . . . 4 β’ (π = π½ β βͺ (π β© π« π₯) = βͺ (π½ β© π« π₯)) |
11 | 8, 10 | mpteq12dv 5197 | . . 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 22387 | . . 3 β’ int = (π β Top β¦ (π₯ β π« βͺ π β¦ βͺ (π β© π« π₯))) | |
13 | 11, 12 | fvmptg 6947 | . 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 686 | 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 1542 β wcel 2107 Vcvv 3444 β© cin 3910 π« cpw 4561 βͺ cuni 4866 β¦ cmpt 5189 βcfv 6497 Topctop 22258 intcnt 22384 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-top 22259 df-ntr 22387 |
This theorem is referenced by: ntrval 22403 ntrrn 42482 ntrf 42483 dssmapntrcls 42488 |
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