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Theorem ussval 24084
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.)
Hypotheses
Ref Expression
ussval.1 𝐵 = (Base‘𝑊)
ussval.2 𝑈 = (UnifSet‘𝑊)
Assertion
Ref Expression
ussval (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Proof of Theorem ussval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ussval.2 . . . 4 𝑈 = (UnifSet‘𝑊)
2 ussval.1 . . . . 5 𝐵 = (Base‘𝑊)
32, 2xpeq12i 5704 . . . 4 (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
41, 3oveq12i 7424 . . 3 (𝑈t (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))
76sqxpeqd 5708 . . . . 5 (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
85, 7oveq12d 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
118, 9, 10fvmpt 6998 . . 3 (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
124, 11eqtr4id 2790 . 2 (𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13 0rest 17382 . . 3 (∅ ↾t (𝐵 × 𝐵)) = ∅
14 fvprc 6883 . . . . 5 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
151, 14eqtrid 2783 . . . 4 𝑊 ∈ V → 𝑈 = ∅)
1615oveq1d 7427 . . 3 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17 fvprc 6883 . . 3 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1813, 16, 173eqtr4a 2797 . 2 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
1912, 18pm2.61i 182 1 (𝑈t (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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