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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 2818 | . . . . . . 7 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imastset 16783 | . . . . . 6 ⊢ (𝜑 → (TopSet‘𝑈) = (𝐽 qTop 𝐹)) |
8 | 5 | fvexi 6677 | . . . . . . 7 ⊢ 𝐽 ∈ V |
9 | fofn 6585 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
11 | fvex 6676 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
12 | 2, 11 | syl6eqel 2918 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V) |
13 | fnex 6971 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑉 ∈ V) → 𝐹 ∈ V) | |
14 | 10, 12, 13 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
15 | eqid 2818 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
16 | 15 | qtopval 22231 | . . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
17 | 8, 14, 16 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
18 | 7, 17 | eqtrd 2853 | . . . . 5 ⊢ (𝜑 → (TopSet‘𝑈) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
19 | ssrab2 4053 | . . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (𝐹 “ ∪ 𝐽) | |
20 | imassrn 5933 | . . . . . . . 8 ⊢ (𝐹 “ ∪ 𝐽) ⊆ ran 𝐹 | |
21 | forn 6586 | . . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
22 | 3, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
23 | 1, 2, 3, 4 | imasbas 16773 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
24 | 22, 23 | eqtrd 2853 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) |
25 | 20, 24 | sseqtrid 4016 | . . . . . . 7 ⊢ (𝜑 → (𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈)) |
26 | sspwb 5332 | . . . . . . 7 ⊢ ((𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈) ↔ 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈)) | |
27 | 25, 26 | sylib 219 | . . . . . 6 ⊢ (𝜑 → 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈)) |
28 | 19, 27 | sstrid 3975 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (Base‘𝑈)) |
29 | 18, 28 | eqsstrd 4002 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈)) |
30 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
31 | 30, 6 | topnid 16697 | . . . 4 ⊢ ((TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈) → (TopSet‘𝑈) = (TopOpen‘𝑈)) |
32 | 29, 31 | syl 17 | . . 3 ⊢ (𝜑 → (TopSet‘𝑈) = (TopOpen‘𝑈)) |
33 | imastopn.o | . . 3 ⊢ 𝑂 = (TopOpen‘𝑈) | |
34 | 32, 33 | syl6eqr 2871 | . 2 ⊢ (𝜑 → (TopSet‘𝑈) = 𝑂) |
35 | 34, 7 | eqtr3d 2855 | 1 ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ◡ccnv 5547 ran crn 5549 “ cima 5551 Fn wfn 6343 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TopSetcts 16559 TopOpenctopn 16683 qTop cqtop 16764 “s cimas 16765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-rest 16684 df-topn 16685 df-qtop 16768 df-imas 16769 |
This theorem is referenced by: imastps 22257 xpstopnlem2 22347 qustgpopn 22655 qustgplem 22656 qustgphaus 22658 imasf1oxms 23026 |
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