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Mirrors > Home > MPE Home > Th. List > resstopn | Structured version Visualization version GIF version |
Description: The topology of a restricted structure. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
resstopn.1 | ⊢ 𝐻 = (𝐾 ↾s 𝐴) |
resstopn.2 | ⊢ 𝐽 = (TopOpen‘𝐾) |
Ref | Expression |
---|---|
resstopn | ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6787 | . . . . 5 ⊢ (TopSet‘𝐾) ∈ V | |
2 | fvex 6787 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
3 | restco 22315 | . . . . 5 ⊢ (((TopSet‘𝐾) ∈ V ∧ (Base‘𝐾) ∈ V ∧ 𝐴 ∈ V) → (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = ((TopSet‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴))) | |
4 | 1, 2, 3 | mp3an12 1450 | . . . 4 ⊢ (𝐴 ∈ V → (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = ((TopSet‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴))) |
5 | resstopn.1 | . . . . . 6 ⊢ 𝐻 = (𝐾 ↾s 𝐴) | |
6 | eqid 2738 | . . . . . 6 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
7 | 5, 6 | resstset 17075 | . . . . 5 ⊢ (𝐴 ∈ V → (TopSet‘𝐾) = (TopSet‘𝐻)) |
8 | incom 4135 | . . . . . 6 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾)) | |
9 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 5, 9 | ressbas 16947 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝐾)) = (Base‘𝐻)) |
11 | 8, 10 | eqtrid 2790 | . . . . 5 ⊢ (𝐴 ∈ V → ((Base‘𝐾) ∩ 𝐴) = (Base‘𝐻)) |
12 | 7, 11 | oveq12d 7293 | . . . 4 ⊢ (𝐴 ∈ V → ((TopSet‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
13 | 4, 12 | eqtrd 2778 | . . 3 ⊢ (𝐴 ∈ V → (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
14 | 9, 6 | topnval 17145 | . . . . 5 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾) |
15 | resstopn.2 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝐾) | |
16 | 14, 15 | eqtr4i 2769 | . . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = 𝐽 |
17 | 16 | oveq1i 7285 | . . 3 ⊢ (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = (𝐽 ↾t 𝐴) |
18 | eqid 2738 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
19 | eqid 2738 | . . . 4 ⊢ (TopSet‘𝐻) = (TopSet‘𝐻) | |
20 | 18, 19 | topnval 17145 | . . 3 ⊢ ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (TopOpen‘𝐻) |
21 | 13, 17, 20 | 3eqtr3g 2801 | . 2 ⊢ (𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
22 | simpr 485 | . . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ∈ V) | |
23 | restfn 17135 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
24 | 23 | fndmi 6537 | . . . . 5 ⊢ dom ↾t = (V × V) |
25 | 24 | ndmov 7456 | . . . 4 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
26 | 22, 25 | nsyl5 159 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = ∅) |
27 | reldmress 16943 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
28 | 27 | ovprc2 7315 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐾 ↾s 𝐴) = ∅) |
29 | 5, 28 | eqtrid 2790 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → 𝐻 = ∅) |
30 | 29 | fveq2d 6778 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = (TopSet‘∅)) |
31 | tsetid 17063 | . . . . . . 7 ⊢ TopSet = Slot (TopSet‘ndx) | |
32 | 31 | str0 16890 | . . . . . 6 ⊢ ∅ = (TopSet‘∅) |
33 | 30, 32 | eqtr4di 2796 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = ∅) |
34 | 33 | oveq1d 7290 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (∅ ↾t (Base‘𝐻))) |
35 | 0rest 17140 | . . . 4 ⊢ (∅ ↾t (Base‘𝐻)) = ∅ | |
36 | 34, 20, 35 | 3eqtr3g 2801 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOpen‘𝐻) = ∅) |
37 | 26, 36 | eqtr4d 2781 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
38 | 21, 37 | pm2.61i 182 | 1 ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∩ cin 3886 ∅c0 4256 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ndxcnx 16894 Basecbs 16912 ↾s cress 16941 TopSetcts 16968 ↾t crest 17131 TopOpenctopn 17132 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-tset 16981 df-rest 17133 df-topn 17134 |
This theorem is referenced by: resstps 22338 submtmd 23255 subgtgp 23256 tsmssubm 23294 invrcn2 23331 ressusp 23416 ressxms 23681 ressms 23682 nrgtdrg 23857 tgioo3 23968 dfii4 24047 retopn 24543 rspectopn 31817 xrge0topn 31893 lmxrge0 31902 qqtopn 31961 |
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