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Theorem ussval 23695
Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6871 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 5698 . . . 4 (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
41, 3oveq12i 7406 . . 3 (𝑈t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
5 fveq2 6879 . . . . 5 (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
6 fveq2 6879 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
76sqxpeqd 5702 . . . . 5 (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
85, 7oveq12d 7412 . . . 4 (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
9 df-uss 23692 . . . 4 UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
10 ovex 7427 . . . 4 ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
118, 9, 10fvmpt 6985 . . 3 (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
124, 11eqtr4id 2791 . 2 (𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13 0rest 17359 . . 3 (∅ ↾t (𝐵 × 𝐵)) = ∅
14 fvprc 6871 . . . . 5 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
151, 14eqtrid 2784 . . . 4 𝑊 ∈ V → 𝑈 = ∅)
1615oveq1d 7409 . . 3 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17 fvprc 6871 . . 3 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1813, 16, 173eqtr4a 2798 . 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 1541  wcel 2106  Vcvv 3474  c0 4319   × cxp 5668  cfv 6533  (class class class)co 7394  Basecbs 17128  UnifSetcunif 17191  t crest 17350  UnifStcuss 23689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-rest 17352  df-uss 23692
This theorem is referenced by:  ussid  23696  ressuss  23698
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