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Mirrors > Home > MPE Home > Th. List > imastopn | Structured version Visualization version GIF version |
Description: The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
imastps.u | β’ (π β π = (πΉ βs π )) |
imastps.v | β’ (π β π = (Baseβπ )) |
imastps.f | β’ (π β πΉ:πβontoβπ΅) |
imastopn.r | β’ (π β π β π) |
imastopn.j | β’ π½ = (TopOpenβπ ) |
imastopn.o | β’ π = (TopOpenβπ) |
Ref | Expression |
---|---|
imastopn | β’ (π β π = (π½ qTop πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imastps.u | . . . . . . 7 β’ (π β π = (πΉ βs π )) | |
2 | imastps.v | . . . . . . 7 β’ (π β π = (Baseβπ )) | |
3 | imastps.f | . . . . . . 7 β’ (π β πΉ:πβontoβπ΅) | |
4 | imastopn.r | . . . . . . 7 β’ (π β π β π) | |
5 | imastopn.j | . . . . . . 7 β’ π½ = (TopOpenβπ ) | |
6 | eqid 2732 | . . . . . . 7 β’ (TopSetβπ) = (TopSetβπ) | |
7 | 1, 2, 3, 4, 5, 6 | imastset 17464 | . . . . . 6 β’ (π β (TopSetβπ) = (π½ qTop πΉ)) |
8 | 5 | fvexi 6902 | . . . . . . 7 β’ π½ β V |
9 | fofn 6804 | . . . . . . . . 9 β’ (πΉ:πβontoβπ΅ β πΉ Fn π) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 β’ (π β πΉ Fn π) |
11 | fvex 6901 | . . . . . . . . 9 β’ (Baseβπ ) β V | |
12 | 2, 11 | eqeltrdi 2841 | . . . . . . . 8 β’ (π β π β V) |
13 | fnex 7215 | . . . . . . . 8 β’ ((πΉ Fn π β§ π β V) β πΉ β V) | |
14 | 10, 12, 13 | syl2anc 584 | . . . . . . 7 β’ (π β πΉ β V) |
15 | eqid 2732 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
16 | 15 | qtopval 23190 | . . . . . . 7 β’ ((π½ β V β§ πΉ β V) β (π½ qTop πΉ) = {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½}) |
17 | 8, 14, 16 | sylancr 587 | . . . . . 6 β’ (π β (π½ qTop πΉ) = {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½}) |
18 | 7, 17 | eqtrd 2772 | . . . . 5 β’ (π β (TopSetβπ) = {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½}) |
19 | ssrab2 4076 | . . . . . 6 β’ {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½} β π« (πΉ β βͺ π½) | |
20 | imassrn 6068 | . . . . . . . 8 β’ (πΉ β βͺ π½) β ran πΉ | |
21 | forn 6805 | . . . . . . . . . 10 β’ (πΉ:πβontoβπ΅ β ran πΉ = π΅) | |
22 | 3, 21 | syl 17 | . . . . . . . . 9 β’ (π β ran πΉ = π΅) |
23 | 1, 2, 3, 4 | imasbas 17454 | . . . . . . . . 9 β’ (π β π΅ = (Baseβπ)) |
24 | 22, 23 | eqtrd 2772 | . . . . . . . 8 β’ (π β ran πΉ = (Baseβπ)) |
25 | 20, 24 | sseqtrid 4033 | . . . . . . 7 β’ (π β (πΉ β βͺ π½) β (Baseβπ)) |
26 | 25 | sspwd 4614 | . . . . . 6 β’ (π β π« (πΉ β βͺ π½) β π« (Baseβπ)) |
27 | 19, 26 | sstrid 3992 | . . . . 5 β’ (π β {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½} β π« (Baseβπ)) |
28 | 18, 27 | eqsstrd 4019 | . . . 4 β’ (π β (TopSetβπ) β π« (Baseβπ)) |
29 | eqid 2732 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
30 | 29, 6 | topnid 17377 | . . . 4 β’ ((TopSetβπ) β π« (Baseβπ) β (TopSetβπ) = (TopOpenβπ)) |
31 | 28, 30 | syl 17 | . . 3 β’ (π β (TopSetβπ) = (TopOpenβπ)) |
32 | imastopn.o | . . 3 β’ π = (TopOpenβπ) | |
33 | 31, 32 | eqtr4di 2790 | . 2 β’ (π β (TopSetβπ) = π) |
34 | 33, 7 | eqtr3d 2774 | 1 β’ (π β π = (π½ qTop πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 {crab 3432 Vcvv 3474 β© cin 3946 β wss 3947 π« cpw 4601 βͺ cuni 4907 β‘ccnv 5674 ran crn 5676 β cima 5678 Fn wfn 6535 βontoβwfo 6538 βcfv 6540 (class class class)co 7405 Basecbs 17140 TopSetcts 17199 TopOpenctopn 17363 qTop cqtop 17445 βs cimas 17446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-rest 17364 df-topn 17365 df-qtop 17449 df-imas 17450 |
This theorem is referenced by: imastps 23216 xpstopnlem2 23306 qustgpopn 23615 qustgplem 23616 qustgphaus 23618 imasf1oxms 23989 |
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