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Theorem ussval 21873
 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.)
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 fveq2 6103 . . . . 5 (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
2 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
32sqxpeqd 5065 . . . . 5 (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
41, 3oveq12d 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
74, 5, 6fvmpt 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)
109, 9xpeq12i 5061 . . . 4 (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
118, 10oveq12i 6561 . . 3 (𝑈t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
127, 11syl6reqr 2663 . 2 (𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13 0rest 15913 . . 3 (∅ ↾t (𝐵 × 𝐵)) = ∅
14 fvprc 6097 . . . . 5 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
158, 14syl5eq 2656 . . . 4 𝑊 ∈ V → 𝑈 = ∅)
1615oveq1d 6564 . . 3 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17 fvprc 6097 . . 3 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1813, 16, 173eqtr4a 2670 . 2 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
1912, 18pm2.61i 175 1 (𝑈t (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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