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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 6689 | . . . . 5 ⊢ (TopSet‘𝐾) ∈ V | |
2 | fvex 6689 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
3 | restco 21917 | . . . . 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 1452 | . . . 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 16770 | . . . . 5 ⊢ (𝐴 ∈ V → (TopSet‘𝐾) = (TopSet‘𝐻)) |
8 | incom 4091 | . . . . . 6 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾)) | |
9 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 5, 9 | ressbas 16659 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝐾)) = (Base‘𝐻)) |
11 | 8, 10 | syl5eq 2785 | . . . . 5 ⊢ (𝐴 ∈ V → ((Base‘𝐾) ∩ 𝐴) = (Base‘𝐻)) |
12 | 7, 11 | oveq12d 7190 | . . . 4 ⊢ (𝐴 ∈ V → ((TopSet‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
13 | 4, 12 | eqtrd 2773 | . . 3 ⊢ (𝐴 ∈ V → (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
14 | 9, 6 | topnval 16813 | . . . . 5 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾) |
15 | resstopn.2 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝐾) | |
16 | 14, 15 | eqtr4i 2764 | . . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = 𝐽 |
17 | 16 | oveq1i 7182 | . . 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 16813 | . . 3 ⊢ ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (TopOpen‘𝐻) |
21 | 13, 17, 20 | 3eqtr3g 2796 | . 2 ⊢ (𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
22 | simpr 488 | . . . 4 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ∈ V) | |
23 | restfn 16803 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
24 | 23 | fndmi 6441 | . . . . 5 ⊢ dom ↾t = (V × V) |
25 | 24 | ndmov 7350 | . . . 4 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
26 | 22, 25 | nsyl5 162 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = ∅) |
27 | reldmress 16655 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
28 | 27 | ovprc2 7212 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐾 ↾s 𝐴) = ∅) |
29 | 5, 28 | syl5eq 2785 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → 𝐻 = ∅) |
30 | 29 | fveq2d 6680 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = (TopSet‘∅)) |
31 | df-tset 16689 | . . . . . . 7 ⊢ TopSet = Slot 9 | |
32 | 31 | str0 16640 | . . . . . 6 ⊢ ∅ = (TopSet‘∅) |
33 | 30, 32 | eqtr4di 2791 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = ∅) |
34 | 33 | oveq1d 7187 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (∅ ↾t (Base‘𝐻))) |
35 | 0rest 16808 | . . . 4 ⊢ (∅ ↾t (Base‘𝐻)) = ∅ | |
36 | 34, 20, 35 | 3eqtr3g 2796 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOpen‘𝐻) = ∅) |
37 | 26, 36 | eqtr4d 2776 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
38 | 21, 37 | pm2.61i 185 | 1 ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∩ cin 3842 ∅c0 4211 × cxp 5523 ‘cfv 6339 (class class class)co 7172 9c9 11780 Basecbs 16588 ↾s cress 16589 TopSetcts 16676 ↾t crest 16799 TopOpenctopn 16800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-tset 16689 df-rest 16801 df-topn 16802 |
This theorem is referenced by: resstps 21940 submtmd 22857 subgtgp 22858 tsmssubm 22896 invrcn2 22933 ressusp 23019 ressxms 23280 ressms 23281 nrgtdrg 23448 tgioo3 23559 dfii4 23638 retopn 24133 rspectopn 31391 xrge0topn 31467 lmxrge0 31476 qqtopn 31533 |
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