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Mirrors > Home > MPE Home > Th. List > ussval | Structured version Visualization version GIF version |
Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6871 to avoid requiring 𝑊 to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
Ref | Expression |
---|---|
ussval.1 | ⊢ 𝐵 = (Base‘𝑊) |
ussval.2 | ⊢ 𝑈 = (UnifSet‘𝑊) |
Ref | Expression |
---|---|
ussval | ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ussval.2 | . . . 4 ⊢ 𝑈 = (UnifSet‘𝑊) | |
2 | ussval.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | 2, 2 | xpeq12i 5698 | . . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊)) |
4 | 1, 3 | oveq12i 7406 | . . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) |
5 | fveq2 6879 | . . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊)) | |
6 | fveq2 6879 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
7 | 6 | sqxpeqd 5702 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊))) |
8 | 5, 7 | oveq12d 7412 | . . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
9 | df-uss 23692 | . . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤)))) | |
10 | ovex 7427 | . . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V | |
11 | 8, 9, 10 | fvmpt 6985 | . . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
12 | 4, 11 | eqtr4id 2791 | . 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)) |
13 | 0rest 17359 | . . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅ | |
14 | fvprc 6871 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅) | |
15 | 1, 14 | eqtrid 2784 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅) |
16 | 15 | oveq1d 7409 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵))) |
17 | fvprc 6871 | . . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅) | |
18 | 13, 16, 17 | 3eqtr4a 2798 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)) |
19 | 12, 18 | pm2.61i 182 | 1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4319 × cxp 5668 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 UnifSetcunif 17191 ↾t crest 17350 UnifStcuss 23689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7959 df-2nd 7960 df-rest 17352 df-uss 23692 |
This theorem is referenced by: ussid 23696 ressuss 23698 |
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