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Theorem ussval 22975
 Description: The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6656 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 5557 . . . 4 (𝐵 × 𝐵) = ((Base‘𝑊) × (Base‘𝑊))
41, 3oveq12i 7169 . . 3 (𝑈t (𝐵 × 𝐵)) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊)))
5 fveq2 6664 . . . . 5 (𝑤 = 𝑊 → (UnifSet‘𝑤) = (UnifSet‘𝑊))
6 fveq2 6664 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
76sqxpeqd 5561 . . . . 5 (𝑤 = 𝑊 → ((Base‘𝑤) × (Base‘𝑤)) = ((Base‘𝑊) × (Base‘𝑊)))
85, 7oveq12d 7175 . . . 4 (𝑤 = 𝑊 → ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
9 df-uss 22972 . . . 4 UnifSt = (𝑤 ∈ V ↦ ((UnifSet‘𝑤) ↾t ((Base‘𝑤) × (Base‘𝑤))))
10 ovex 7190 . . . 4 ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))) ∈ V
118, 9, 10fvmpt 6765 . . 3 (𝑊 ∈ V → (UnifSt‘𝑊) = ((UnifSet‘𝑊) ↾t ((Base‘𝑊) × (Base‘𝑊))))
124, 11eqtr4id 2813 . 2 (𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
13 0rest 16776 . . 3 (∅ ↾t (𝐵 × 𝐵)) = ∅
14 fvprc 6656 . . . . 5 𝑊 ∈ V → (UnifSet‘𝑊) = ∅)
151, 14syl5eq 2806 . . . 4 𝑊 ∈ V → 𝑈 = ∅)
1615oveq1d 7172 . . 3 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (∅ ↾t (𝐵 × 𝐵)))
17 fvprc 6656 . . 3 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1813, 16, 173eqtr4a 2820 . 2 𝑊 ∈ V → (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊))
1912, 18pm2.61i 185 1 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ∅c0 4228   × cxp 5527  ‘cfv 6341  (class class class)co 7157  Basecbs 16556  UnifSetcunif 16648   ↾t crest 16767  UnifStcuss 22969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-rest 16769  df-uss 22972 This theorem is referenced by:  ussid  22976  ressuss  22979
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