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Theorem ussid 24158
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 7422 . . 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 6905 . . . . . . 7 𝐵 ∈ V
54, 4xpex 7749 . . . . . 6 (𝐵 × 𝐵) ∈ V
62, 5eqeltrrdi 2837 . . . . 5 ((𝐵 × 𝐵) = 𝑈 𝑈 ∈ V)
7 uniexb 7760 . . . . 5 (𝑈 ∈ V ↔ 𝑈 ∈ V)
86, 7sylibr 233 . . . 4 ((𝐵 × 𝐵) = 𝑈𝑈 ∈ V)
9 eqid 2727 . . . . 5 𝑈 = 𝑈
109restid 17408 . . . 4 (𝑈 ∈ V → (𝑈t 𝑈) = 𝑈)
118, 10syl 17 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t 𝑈) = 𝑈)
121, 11eqtr2d 2768 . 2 ((𝐵 × 𝐵) = 𝑈𝑈 = (𝑈t (𝐵 × 𝐵)))
13 ussval.2 . . 3 𝑈 = (UnifSet‘𝑊)
143, 13ussval 24157 . 2 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
1512, 14eqtrdi 2783 1 ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3469   cuni 4903   × cxp 5670  cfv 6542  (class class class)co 7414  Basecbs 17173  UnifSetcunif 17236  t crest 17395  UnifStcuss 24151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-rest 17397  df-uss 24154
This theorem is referenced by:  tususs  24168  cnflduss  25277
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