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Mirrors > Home > MPE Home > Th. List > ressuss | Structured version Visualization version GIF version |
Description: Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.) |
Ref | Expression |
---|---|
ressuss | ⊢ (𝐴 ∈ 𝑉 → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2778 | . . . . 5 ⊢ (UnifSet‘𝑊) = (UnifSet‘𝑊) | |
3 | 1, 2 | ussval 22471 | . . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) = (UnifSt‘𝑊) |
4 | 3 | oveq1i 6932 | . . 3 ⊢ (((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ↾t (𝐴 × 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) |
5 | fvex 6459 | . . . . 5 ⊢ (UnifSet‘𝑊) ∈ V | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (UnifSet‘𝑊) ∈ V) |
7 | fvex 6459 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
8 | 7, 7 | xpex 7240 | . . . . 5 ⊢ ((Base‘𝑊) × (Base‘𝑊)) ∈ V |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Base‘𝑊) × (Base‘𝑊)) ∈ V) |
10 | sqxpexg 7241 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
11 | restco 21376 | . . . 4 ⊢ (((UnifSet‘𝑊) ∈ V ∧ ((Base‘𝑊) × (Base‘𝑊)) ∈ V ∧ (𝐴 × 𝐴) ∈ V) → (((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ↾t (𝐴 × 𝐴)) = ((UnifSet‘𝑊) ↾t (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴)))) | |
12 | 6, 9, 10, 11 | syl3anc 1439 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ↾t (𝐴 × 𝐴)) = ((UnifSet‘𝑊) ↾t (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴)))) |
13 | 4, 12 | syl5eqr 2828 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)) = ((UnifSet‘𝑊) ↾t (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴)))) |
14 | inxp 5500 | . . . . 5 ⊢ (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴)) = (((Base‘𝑊) ∩ 𝐴) × ((Base‘𝑊) ∩ 𝐴)) | |
15 | incom 4028 | . . . . . . 7 ⊢ (𝐴 ∩ (Base‘𝑊)) = ((Base‘𝑊) ∩ 𝐴) | |
16 | eqid 2778 | . . . . . . . 8 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴) | |
17 | 16, 1 | ressbas 16326 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴))) |
18 | 15, 17 | syl5eqr 2828 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((Base‘𝑊) ∩ 𝐴) = (Base‘(𝑊 ↾s 𝐴))) |
19 | 18 | sqxpeqd 5387 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (((Base‘𝑊) ∩ 𝐴) × ((Base‘𝑊) ∩ 𝐴)) = ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴)))) |
20 | 14, 19 | syl5eq 2826 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴)) = ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴)))) |
21 | 20 | oveq2d 6938 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((UnifSet‘𝑊) ↾t (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴))) = ((UnifSet‘𝑊) ↾t ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴))))) |
22 | 16, 2 | ressunif 22474 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (UnifSet‘𝑊) = (UnifSet‘(𝑊 ↾s 𝐴))) |
23 | 22 | oveq1d 6937 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((UnifSet‘𝑊) ↾t ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴)))) = ((UnifSet‘(𝑊 ↾s 𝐴)) ↾t ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴))))) |
24 | eqid 2778 | . . . . 5 ⊢ (Base‘(𝑊 ↾s 𝐴)) = (Base‘(𝑊 ↾s 𝐴)) | |
25 | eqid 2778 | . . . . 5 ⊢ (UnifSet‘(𝑊 ↾s 𝐴)) = (UnifSet‘(𝑊 ↾s 𝐴)) | |
26 | 24, 25 | ussval 22471 | . . . 4 ⊢ ((UnifSet‘(𝑊 ↾s 𝐴)) ↾t ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴)))) = (UnifSt‘(𝑊 ↾s 𝐴)) |
27 | 26 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((UnifSet‘(𝑊 ↾s 𝐴)) ↾t ((Base‘(𝑊 ↾s 𝐴)) × (Base‘(𝑊 ↾s 𝐴)))) = (UnifSt‘(𝑊 ↾s 𝐴))) |
28 | 21, 23, 27 | 3eqtrd 2818 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((UnifSet‘𝑊) ↾t (((Base‘𝑊) × (Base‘𝑊)) ∩ (𝐴 × 𝐴))) = (UnifSt‘(𝑊 ↾s 𝐴))) |
29 | 13, 28 | eqtr2d 2815 | 1 ⊢ (𝐴 ∈ 𝑉 → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∩ cin 3791 × cxp 5353 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 ↾s cress 16256 UnifSetcunif 16348 ↾t crest 16467 UnifStcuss 22465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-unif 16361 df-rest 16469 df-uss 22468 |
This theorem is referenced by: ressust 22476 ressusp 22477 ucnextcn 22516 reust 23587 qqhucn 30634 |
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