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Theorem ussid 22111
Description: In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
ussval.1 𝐵 = (Base‘𝑊)
ussval.2 𝑈 = (UnifSet‘𝑊)
Assertion
Ref Expression
ussid ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))

Proof of Theorem ussid
StepHypRef Expression
1 oveq2 6698 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t (𝐵 × 𝐵)) = (𝑈t 𝑈))
2 id 22 . . . . . 6 ((𝐵 × 𝐵) = 𝑈 → (𝐵 × 𝐵) = 𝑈)
3 ussval.1 . . . . . . . 8 𝐵 = (Base‘𝑊)
4 fvex 6239 . . . . . . . 8 (Base‘𝑊) ∈ V
53, 4eqeltri 2726 . . . . . . 7 𝐵 ∈ V
65, 5xpex 7004 . . . . . 6 (𝐵 × 𝐵) ∈ V
72, 6syl6eqelr 2739 . . . . 5 ((𝐵 × 𝐵) = 𝑈 𝑈 ∈ V)
8 uniexb 7015 . . . . 5 (𝑈 ∈ V ↔ 𝑈 ∈ V)
97, 8sylibr 224 . . . 4 ((𝐵 × 𝐵) = 𝑈𝑈 ∈ V)
10 eqid 2651 . . . . 5 𝑈 = 𝑈
1110restid 16141 . . . 4 (𝑈 ∈ V → (𝑈t 𝑈) = 𝑈)
129, 11syl 17 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t 𝑈) = 𝑈)
131, 12eqtr2d 2686 . 2 ((𝐵 × 𝐵) = 𝑈𝑈 = (𝑈t (𝐵 × 𝐵)))
14 ussval.2 . . 3 𝑈 = (UnifSet‘𝑊)
153, 14ussval 22110 . 2 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
1613, 15syl6eq 2701 1 ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231   cuni 4468   × cxp 5141  cfv 5926  (class class class)co 6690  Basecbs 15904  UnifSetcunif 15998  t crest 16128  UnifStcuss 22104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-rest 16130  df-uss 22107
This theorem is referenced by:  tususs  22121  cnflduss  23198
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