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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 6883 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 5704 | . . . 4 ⊢ (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊)) |
4 | 1, 3 | oveq12i 7424 | . . 3 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) |
5 | fveq2 6891 | . . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊)) | |
6 | fveq2 6891 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
7 | 6 | sqxpeqd 5708 | . . . . 5 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊))) |
8 | 5, 7 | oveq12d 7430 | . . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
9 | df-uss 24081 | . . . 4 ⊢ UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤)))) | |
10 | ovex 7445 | . . . 4 ⊢ ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V | |
11 | 8, 9, 10 | fvmpt 6998 | . . 3 ⊢ (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))) |
12 | 4, 11 | eqtr4id 2790 | . 2 ⊢ (𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)) |
13 | 0rest 17382 | . . 3 ⊢ (∅ ↾t (𝐵 × 𝐵)) = ∅ | |
14 | fvprc 6883 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (UnifSet‘𝑊) = ∅) | |
15 | 1, 14 | eqtrid 2783 | . . . 4 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅) |
16 | 15 | oveq1d 7427 | . . 3 ⊢ (¬ 𝑊 ∈ V → (𝑈 ↾t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵))) |
17 | fvprc 6883 | . . 3 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅) | |
18 | 13, 16, 17 | 3eqtr4a 2797 | . 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 1540 ∈ wcel 2105 Vcvv 3473 ∅c0 4322 × cxp 5674 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 UnifSetcunif 17214 ↾t crest 17373 UnifStcuss 24078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-rest 17375 df-uss 24081 |
This theorem is referenced by: ussid 24085 ressuss 24087 |
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