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Theorem ussval 21976
Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6144 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 6150 . . . . 5 (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
2 fveq2 6150 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
32sqxpeqd 5103 . . . . 5 (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
41, 3oveq12d 6625 . . . 4 (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
5 df-uss 21973 . . . 4 UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
6 ovex 6635 . . . 4 ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
74, 5, 6fvmpt 6241 . . 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 5099 . . . 4 (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
118, 10oveq12i 6619 . . 3 (𝑈t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
127, 11syl6reqr 2674 . 2 (𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13 0rest 16014 . . 3 (∅ ↾t (𝐵 × 𝐵)) = ∅
14 fvprc 6144 . . . . 5 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
158, 14syl5eq 2667 . . . 4 𝑊 ∈ V → 𝑈 = ∅)
1615oveq1d 6622 . . 3 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17 fvprc 6144 . . 3 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1813, 16, 173eqtr4a 2681 . 2 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
1912, 18pm2.61i 176 1 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3186  c0 3893   × cxp 5074  cfv 5849  (class class class)co 6607  Basecbs 15784  UnifSetcunif 15875  t crest 16005  UnifStcuss 21970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-rest 16007  df-uss 21973
This theorem is referenced by:  ussid  21977  ressuss  21980
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