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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 16628 | . . . . . 6 ⊢ (𝜑 → (TopSet‘𝑈) = (𝐽 qTop 𝐹)) |
8 | 5 | fvexi 6559 | . . . . . . 7 ⊢ 𝐽 ∈ V |
9 | fofn 6467 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
11 | fvex 6558 | . . . . . . . . 9 ⊢ (Base‘𝑅) ∈ V | |
12 | 2, 11 | syl6eqel 2893 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V) |
13 | fnex 6853 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑉 ∈ V) → 𝐹 ∈ V) | |
14 | 10, 12, 13 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
15 | eqid 2797 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
16 | 15 | qtopval 21991 | . . . . . . 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 2833 | . . . . 5 ⊢ (𝜑 → (TopSet‘𝑈) = {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽}) |
19 | ssrab2 3983 | . . . . . 6 ⊢ {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (𝐹 “ ∪ 𝐽) | |
20 | imassrn 5824 | . . . . . . . 8 ⊢ (𝐹 “ ∪ 𝐽) ⊆ ran 𝐹 | |
21 | forn 6468 | . . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
22 | 3, 21 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
23 | 1, 2, 3, 4 | imasbas 16618 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
24 | 22, 23 | eqtrd 2833 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑈)) |
25 | 20, 24 | sseqtrid 3946 | . . . . . . 7 ⊢ (𝜑 → (𝐹 “ ∪ 𝐽) ⊆ (Base‘𝑈)) |
26 | sspwb 5240 | . . . . . . 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 3906 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ 𝒫 (𝐹 “ ∪ 𝐽) ∣ ((◡𝐹 “ 𝑥) ∩ ∪ 𝐽) ∈ 𝐽} ⊆ 𝒫 (Base‘𝑈)) |
29 | 18, 28 | eqsstrd 3932 | . . . 4 ⊢ (𝜑 → (TopSet‘𝑈) ⊆ 𝒫 (Base‘𝑈)) |
30 | eqid 2797 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
31 | 30, 6 | topnid 16542 | . . . 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 2851 | . 2 ⊢ (𝜑 → (TopSet‘𝑈) = 𝑂) |
35 | 34, 7 | eqtr3d 2835 | 1 ⊢ (𝜑 → 𝑂 = (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 {crab 3111 Vcvv 3440 ∩ cin 3864 ⊆ wss 3865 𝒫 cpw 4459 ∪ cuni 4751 ◡ccnv 5449 ran crn 5451 “ cima 5453 Fn wfn 6227 –onto→wfo 6230 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 TopSetcts 16404 TopOpenctopn 16528 qTop cqtop 16609 “s cimas 16610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-rest 16529 df-topn 16530 df-qtop 16613 df-imas 16614 |
This theorem is referenced by: imastps 22017 xpstopnlem2 22107 qustgpopn 22415 qustgplem 22416 qustgphaus 22418 imasf1oxms 22786 |
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