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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrrn | Structured version Visualization version GIF version |
Description: The range of the interior function of a topology a subset of 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 |
---|---|
ntrrn | β’ (π½ β Top β ran πΌ β π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrrn.i | . . 3 β’ πΌ = (intβπ½) | |
2 | 1 | rneqi 5896 | . 2 β’ ran πΌ = ran (intβπ½) |
3 | vpwex 5336 | . . . . . . . 8 β’ π« π β V | |
4 | 3 | inex2 5279 | . . . . . . 7 β’ (π½ β© π« π ) β V |
5 | 4 | uniex 7682 | . . . . . 6 β’ βͺ (π½ β© π« π ) β V |
6 | 5 | rgenw 3065 | . . . . 5 β’ βπ β π« πβͺ (π½ β© π« π ) β V |
7 | nfcv 2904 | . . . . . 6 β’ β²π π« π | |
8 | 7 | fnmptf 6641 | . . . . 5 β’ (βπ β π« πβͺ (π½ β© π« π ) β V β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π) |
9 | 6, 8 | mp1i 13 | . . . 4 β’ (π½ β Top β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π) |
10 | ntrrn.x | . . . . . 6 β’ π = βͺ π½ | |
11 | 10 | ntrfval 22398 | . . . . 5 β’ (π½ β Top β (intβπ½) = (π β π« π β¦ βͺ (π½ β© π« π ))) |
12 | 11 | fneq1d 6599 | . . . 4 β’ (π½ β Top β ((intβπ½) Fn π« π β (π β π« π β¦ βͺ (π½ β© π« π )) Fn π« π)) |
13 | 9, 12 | mpbird 257 | . . 3 β’ (π½ β Top β (intβπ½) Fn π« π) |
14 | elpwi 4571 | . . . . 5 β’ (π β π« π β π β π) | |
15 | 10 | ntropn 22423 | . . . . . 6 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ ) β π½) |
16 | 15 | ex 414 | . . . . 5 β’ (π½ β Top β (π β π β ((intβπ½)βπ ) β π½)) |
17 | 14, 16 | syl5 34 | . . . 4 β’ (π½ β Top β (π β π« π β ((intβπ½)βπ ) β π½)) |
18 | 17 | ralrimiv 3139 | . . 3 β’ (π½ β Top β βπ β π« π((intβπ½)βπ ) β π½) |
19 | fnfvrnss 7072 | . . 3 β’ (((intβπ½) Fn π« π β§ βπ β π« π((intβπ½)βπ ) β π½) β ran (intβπ½) β π½) | |
20 | 13, 18, 19 | syl2anc 585 | . 2 β’ (π½ β Top β ran (intβπ½) β π½) |
21 | 2, 20 | eqsstrid 3996 | 1 β’ (π½ β Top β ran πΌ β π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 β© cin 3913 β wss 3914 π« cpw 4564 βͺ cuni 4869 β¦ cmpt 5192 ran crn 5638 Fn wfn 6495 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: ntrf 42487 |
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