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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 6113 | . . . . 5 ⊢ (TopSet‘𝐾) ∈ V | |
2 | fvex 6113 | . . . . 5 ⊢ (Base‘𝐾) ∈ V | |
3 | restco 20778 | . . . . 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 1406 | . . . 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 2610 | . . . . . 6 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾) | |
7 | 5, 6 | resstset 15869 | . . . . 5 ⊢ (𝐴 ∈ V → (TopSet‘𝐾) = (TopSet‘𝐻)) |
8 | incom 3767 | . . . . . 6 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾)) | |
9 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 5, 9 | ressbas 15757 | . . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 ∩ (Base‘𝐾)) = (Base‘𝐻)) |
11 | 8, 10 | syl5eq 2656 | . . . . 5 ⊢ (𝐴 ∈ V → ((Base‘𝐾) ∩ 𝐴) = (Base‘𝐻)) |
12 | 7, 11 | oveq12d 6567 | . . . 4 ⊢ (𝐴 ∈ V → ((TopSet‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
13 | 4, 12 | eqtrd 2644 | . . 3 ⊢ (𝐴 ∈ V → (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = ((TopSet‘𝐻) ↾t (Base‘𝐻))) |
14 | 9, 6 | topnval 15918 | . . . . 5 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾) |
15 | resstopn.2 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝐾) | |
16 | 14, 15 | eqtr4i 2635 | . . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = 𝐽 |
17 | 16 | oveq1i 6559 | . . 3 ⊢ (((TopSet‘𝐾) ↾t (Base‘𝐾)) ↾t 𝐴) = (𝐽 ↾t 𝐴) |
18 | eqid 2610 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
19 | eqid 2610 | . . . 4 ⊢ (TopSet‘𝐻) = (TopSet‘𝐻) | |
20 | 18, 19 | topnval 15918 | . . 3 ⊢ ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (TopOpen‘𝐻) |
21 | 13, 17, 20 | 3eqtr3g 2667 | . 2 ⊢ (𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
22 | simpr 476 | . . . . 5 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → 𝐴 ∈ V) | |
23 | 22 | con3i 149 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
24 | restfn 15908 | . . . . . 6 ⊢ ↾t Fn (V × V) | |
25 | fndm 5904 | . . . . . 6 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
26 | 24, 25 | ax-mp 5 | . . . . 5 ⊢ dom ↾t = (V × V) |
27 | 26 | ndmov 6716 | . . . 4 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
28 | 23, 27 | syl 17 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = ∅) |
29 | reldmress 15753 | . . . . . . . . 9 ⊢ Rel dom ↾s | |
30 | 29 | ovprc2 6583 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → (𝐾 ↾s 𝐴) = ∅) |
31 | 5, 30 | syl5eq 2656 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → 𝐻 = ∅) |
32 | 31 | fveq2d 6107 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = (TopSet‘∅)) |
33 | df-tset 15787 | . . . . . . 7 ⊢ TopSet = Slot 9 | |
34 | 33 | str0 15739 | . . . . . 6 ⊢ ∅ = (TopSet‘∅) |
35 | 32, 34 | syl6eqr 2662 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (TopSet‘𝐻) = ∅) |
36 | 35 | oveq1d 6564 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ((TopSet‘𝐻) ↾t (Base‘𝐻)) = (∅ ↾t (Base‘𝐻))) |
37 | 0rest 15913 | . . . 4 ⊢ (∅ ↾t (Base‘𝐻)) = ∅ | |
38 | 36, 20, 37 | 3eqtr3g 2667 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOpen‘𝐻) = ∅) |
39 | 28, 38 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ V → (𝐽 ↾t 𝐴) = (TopOpen‘𝐻)) |
40 | 21, 39 | pm2.61i 175 | 1 ⊢ (𝐽 ↾t 𝐴) = (TopOpen‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ∅c0 3874 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 9c9 10954 Basecbs 15695 ↾s cress 15696 TopSetcts 15774 ↾t crest 15904 TopOpenctopn 15905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-tset 15787 df-rest 15906 df-topn 15907 |
This theorem is referenced by: resstps 20801 submtmd 21718 subgtgp 21719 tsmssubm 21756 invrcn2 21793 ressusp 21879 ressxms 22140 ressms 22141 nrgtdrg 22307 tgioo3 22416 dfii4 22495 retopn 22975 xrge0topn 29317 lmxrge0 29326 qqtopn 29383 |
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