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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 2797 | . . . . . . 7 ⊢ (TopSet‘𝑈) = (TopSet‘𝑈) | |
7 | 1, 2, 3, 4, 5, 6 | imastset 16493 | . . . . . 6 ⊢ (𝜑 → (TopSet‘𝑈) = (𝐽 qTop 𝐹)) |
8 | 5 | fvexi 6423 | . . . . . . 7 ⊢ 𝐽 ∈ V |
9 | fofn 6331 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
11 | fvex 6422 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
12 | 2, 11 | syl6eqel 2884 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V) |
13 | fnex 6708 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑉 ∈ V) → 𝐹 ∈ V) | |
14 | 10, 12, 13 | syl2anc 580 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
15 | eqid 2797 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
16 | 15 | qtopval 21823 | . . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝐹 ∈ V) → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
17 | 8, 14, 16 | sylancr 582 | . . . . . 6 ⊢ (𝜑 → (𝐽 qTop 𝐹) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
18 | 7, 17 | eqtrd 2831 | . . . . 5 ⊢ (𝜑 → (TopSet‘𝑈) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
19 | ssrab2 3881 | . . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (𝐹 “ ∪ 𝐽) | |
20 | imassrn 5692 | . . . . . . . 8 ⊢ (𝐹 “ ∪ 𝐽) ⊆ ran 𝐹 | |
21 | forn 6332 | . . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
22 | 3, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
23 | 1, 2, 3, 4 | imasbas 16483 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
24 | 22, 23 | eqtrd 2831 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) |
25 | 20, 24 | syl5sseq 3847 | . . . . . . 7 ⊢ (𝜑 → (𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈)) |
26 | sspwb 5106 | . . . . . . 7 ⊢ ((𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈) ↔ 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈)) | |
27 | 25, 26 | sylib 210 | . . . . . 6 ⊢ (𝜑 → 𝒫 (𝐹 “ ∪ 𝐽) ⊆ 𝒫 (Base‘𝑈)) |
28 | 19, 27 | syl5ss 3807 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (Base‘𝑈)) |
29 | 18, 28 | eqsstrd 3833 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈)) |
30 | eqid 2797 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
31 | 30, 6 | topnid 16407 | . . . 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 2849 | . 2 ⊢ (𝜑 → (TopSet‘𝑈) = 𝑂) |
35 | 34, 7 | eqtr3d 2833 | 1 ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {crab 3091 Vcvv 3383 ∩ cin 3766 ⊆ wss 3767 𝒫 cpw 4347 ∪ cuni 4626 ◡ccnv 5309 ran crn 5311 “ cima 5313 Fn wfn 6094 –onto→wfo 6097 ‘cfv 6099 (class class class)co 6876 Basecbs 16180 TopSetcts 16269 TopOpenctopn 16393 qTop cqtop 16474 “s cimas 16475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-struct 16182 df-ndx 16183 df-slot 16184 df-base 16186 df-plusg 16276 df-mulr 16277 df-sca 16279 df-vsca 16280 df-ip 16281 df-tset 16282 df-ple 16283 df-ds 16285 df-rest 16394 df-topn 16395 df-qtop 16478 df-imas 16479 |
This theorem is referenced by: imastps 21849 xpstopnlem2 21939 qustgpopn 22247 qustgplem 22248 qustgphaus 22250 imasf1oxms 22618 |
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