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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 6097 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 | fveq2 6103 | . . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊)) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
3 | 2 | sqxpeqd 5065 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊))) |
4 | 1, 3 | oveq12d 6567 | . . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
5 | df-uss 21870 | . . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤)))) | |
6 | ovex 6577 | . . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6191 | . . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
8 | ussval.2 | . . . 4 ⊢ 𝑈 = (UnifSet‘𝑊) | |
9 | ussval.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
10 | 9, 9 | xpeq12i 5061 | . . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊)) |
11 | 8, 10 | oveq12i 6561 | . . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) |
12 | 7, 11 | syl6reqr 2663 | . 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)) |
13 | 0rest 15913 | . . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅ | |
14 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅) | |
15 | 8, 14 | syl5eq 2656 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅) |
16 | 15 | oveq1d 6564 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵))) |
17 | fvprc 6097 | . . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅) | |
18 | 13, 16, 17 | 3eqtr4a 2670 | . 2 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)) |
19 | 12, 18 | pm2.61i 175 | 1 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 UnifSetcunif 15778 ↾t crest 15904 UnifStcuss 21867 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-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-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 df-uss 21870 |
This theorem is referenced by: ussid 21874 ressuss 21877 |
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