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Theorem ussid 24269
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 7439 . . 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 6920 . . . . . . 7 𝐵 ∈ V
54, 4xpex 7773 . . . . . 6 (𝐵 × 𝐵) ∈ V
62, 5eqeltrrdi 2850 . . . . 5 ((𝐵 × 𝐵) = 𝑈 𝑈 ∈ V)
7 uniexb 7784 . . . . 5 (𝑈 ∈ V ↔ 𝑈 ∈ V)
86, 7sylibr 234 . . . 4 ((𝐵 × 𝐵) = 𝑈𝑈 ∈ V)
9 eqid 2737 . . . . 5 𝑈 = 𝑈
109restid 17478 . . . 4 (𝑈 ∈ V → (𝑈t 𝑈) = 𝑈)
118, 10syl 17 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t 𝑈) = 𝑈)
121, 11eqtr2d 2778 . 2 ((𝐵 × 𝐵) = 𝑈𝑈 = (𝑈t (𝐵 × 𝐵)))
13 ussval.2 . . 3 𝑈 = (UnifSet‘𝑊)
143, 13ussval 24268 . 2 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
1512, 14eqtrdi 2793 1 ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480   cuni 4907   × cxp 5683  cfv 6561  (class class class)co 7431  Basecbs 17247  UnifSetcunif 17307  t crest 17465  UnifStcuss 24262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467  df-uss 24265
This theorem is referenced by:  tususs  24279  cnflduss  25390
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