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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 | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imastps.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imastps.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imastopn.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
imastopn.j | ⊢ 𝐽 = (TopOpen‘𝑅) |
imastopn.o | ⊢ 𝑂 = (TopOpen‘𝑈) |
Ref | Expression |
---|---|
imastopn | ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imastps.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imastps.v | . . . . . . 7 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imastps.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
4 | imastopn.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
5 | imastopn.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝑅) | |
6 | eqid 2821 | . . . . . . 7 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imastset 16794 | . . . . . 6 ⊢ (𝜑 → (TopSet‘𝑈) = (𝐽 qTop 𝐹)) |
8 | 5 | fvexi 6683 | . . . . . . 7 ⊢ 𝐽 ∈ V |
9 | fofn 6591 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
11 | fvex 6682 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
12 | 2, 11 | eqeltrdi 2921 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V) |
13 | fnex 6979 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑉 ∈ V) → 𝐹 ∈ V) | |
14 | 10, 12, 13 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
15 | eqid 2821 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
16 | 15 | qtopval 22302 | . . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
17 | 8, 14, 16 | sylancr 589 | . . . . . 6 ⊢ (𝜑 → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
18 | 7, 17 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (TopSet‘𝑈) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
19 | ssrab2 4055 | . . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (𝐹 “ ∪ 𝐽) | |
20 | imassrn 5939 | . . . . . . . 8 ⊢ (𝐹 “ ∪ 𝐽) ⊆ ran 𝐹 | |
21 | forn 6592 | . . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
22 | 3, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
23 | 1, 2, 3, 4 | imasbas 16784 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
24 | 22, 23 | eqtrd 2856 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) |
25 | 20, 24 | sseqtrid 4018 | . . . . . . 7 ⊢ (𝜑 → (𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈)) |
26 | 25 | sspwd 4553 | . . . . . 6 ⊢ (𝜑 → 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈)) |
27 | 19, 26 | sstrid 3977 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (Base‘𝑈)) |
28 | 18, 27 | eqsstrd 4004 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈)) |
29 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
30 | 29, 6 | topnid 16708 | . . . 4 ⊢ ((TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈) → (TopSet‘𝑈) = (TopOpen‘𝑈)) |
31 | 28, 30 | syl 17 | . . 3 ⊢ (𝜑 → (TopSet‘𝑈) = (TopOpen‘𝑈)) |
32 | imastopn.o | . . 3 ⊢ 𝑂 = (TopOpen‘𝑈) | |
33 | 31, 32 | syl6eqr 2874 | . 2 ⊢ (𝜑 → (TopSet‘𝑈) = 𝑂) |
34 | 33, 7 | eqtr3d 2858 | 1 ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4837 ◡ccnv 5553 ran crn 5555 “ cima 5557 Fn wfn 6349 –onto→wfo 6352 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 TopSetcts 16570 TopOpenctopn 16694 qTop cqtop 16775 “s cimas 16776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-rest 16695 df-topn 16696 df-qtop 16779 df-imas 16780 |
This theorem is referenced by: imastps 22328 xpstopnlem2 22418 qustgpopn 22727 qustgplem 22728 qustgphaus 22730 imasf1oxms 23098 |
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