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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 2733 | . . . . . . 7 β’ (TopSetβπ) = (TopSetβπ) | |
7 | 1, 2, 3, 4, 5, 6 | imastset 17409 | . . . . . 6 β’ (π β (TopSetβπ) = (π½ qTop πΉ)) |
8 | 5 | fvexi 6857 | . . . . . . 7 β’ π½ β V |
9 | fofn 6759 | . . . . . . . . 9 β’ (πΉ:πβontoβπ΅ β πΉ Fn π) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 β’ (π β πΉ Fn π) |
11 | fvex 6856 | . . . . . . . . 9 β’ (Baseβπ ) β V | |
12 | 2, 11 | eqeltrdi 2842 | . . . . . . . 8 β’ (π β π β V) |
13 | fnex 7168 | . . . . . . . 8 β’ ((πΉ Fn π β§ π β V) β πΉ β V) | |
14 | 10, 12, 13 | syl2anc 585 | . . . . . . 7 β’ (π β πΉ β V) |
15 | eqid 2733 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
16 | 15 | qtopval 23062 | . . . . . . 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 588 | . . . . . 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 2773 | . . . . 5 β’ (π β (TopSetβπ) = {π₯ β π« (πΉ β βͺ π½) β£ ((β‘πΉ β π₯) β© βͺ π½) β π½}) |
19 | ssrab2 4038 | . . . . . 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 6025 | . . . . . . . 8 β’ (πΉ β βͺ π½) β ran πΉ | |
21 | forn 6760 | . . . . . . . . . 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 17399 | . . . . . . . . 9 β’ (π β π΅ = (Baseβπ)) |
24 | 22, 23 | eqtrd 2773 | . . . . . . . 8 β’ (π β ran πΉ = (Baseβπ)) |
25 | 20, 24 | sseqtrid 3997 | . . . . . . 7 β’ (π β (πΉ β βͺ π½) β (Baseβπ)) |
26 | 25 | sspwd 4574 | . . . . . 6 β’ (π β π« (πΉ β βͺ π½) β π« (Baseβπ)) |
27 | 19, 26 | sstrid 3956 | . . . . 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 3983 | . . . 4 β’ (π β (TopSetβπ) β π« (Baseβπ)) |
29 | eqid 2733 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
30 | 29, 6 | topnid 17322 | . . . 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 2791 | . 2 β’ (π β (TopSetβπ) = π) |
34 | 33, 7 | eqtr3d 2775 | 1 β’ (π β π = (π½ qTop πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 β© cin 3910 β wss 3911 π« cpw 4561 βͺ cuni 4866 β‘ccnv 5633 ran crn 5635 β cima 5637 Fn wfn 6492 βontoβwfo 6495 βcfv 6497 (class class class)co 7358 Basecbs 17088 TopSetcts 17144 TopOpenctopn 17308 qTop cqtop 17390 βs cimas 17391 |
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 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 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-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 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-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-rest 17309 df-topn 17310 df-qtop 17394 df-imas 17395 |
This theorem is referenced by: imastps 23088 xpstopnlem2 23178 qustgpopn 23487 qustgplem 23488 qustgphaus 23490 imasf1oxms 23861 |
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