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Theorem ussid 22472
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 6930 . . 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)
43fvexi 6460 . . . . . . 7 𝐵 ∈ V
54, 4xpex 7240 . . . . . 6 (𝐵 × 𝐵) ∈ V
62, 5syl6eqelr 2867 . . . . 5 ((𝐵 × 𝐵) = 𝑈 𝑈 ∈ V)
7 uniexb 7250 . . . . 5 (𝑈 ∈ V ↔ 𝑈 ∈ V)
86, 7sylibr 226 . . . 4 ((𝐵 × 𝐵) = 𝑈𝑈 ∈ V)
9 eqid 2777 . . . . 5 𝑈 = 𝑈
109restid 16480 . . . 4 (𝑈 ∈ V → (𝑈t 𝑈) = 𝑈)
118, 10syl 17 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t 𝑈) = 𝑈)
121, 11eqtr2d 2814 . 2 ((𝐵 × 𝐵) = 𝑈𝑈 = (𝑈t (𝐵 × 𝐵)))
13 ussval.2 . . 3 𝑈 = (UnifSet‘𝑊)
143, 13ussval 22471 . 2 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
1512, 14syl6eq 2829 1 ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  Vcvv 3397   cuni 4671   × cxp 5353  cfv 6135  (class class class)co 6922  Basecbs 16255  UnifSetcunif 16348  t crest 16467  UnifStcuss 22465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-rest 16469  df-uss 22468
This theorem is referenced by:  tususs  22482  cnflduss  23562
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