Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ussid Structured version   Visualization version   GIF version

Theorem ussid 22864
 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 7154 . . 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 6673 . . . . . . 7 𝐵 ∈ V
54, 4xpex 7467 . . . . . 6 (𝐵 × 𝐵) ∈ V
62, 5eqeltrrdi 2925 . . . . 5 ((𝐵 × 𝐵) = 𝑈 𝑈 ∈ V)
7 uniexb 7477 . . . . 5 (𝑈 ∈ V ↔ 𝑈 ∈ V)
86, 7sylibr 237 . . . 4 ((𝐵 × 𝐵) = 𝑈𝑈 ∈ V)
9 eqid 2824 . . . . 5 𝑈 = 𝑈
109restid 16705 . . . 4 (𝑈 ∈ V → (𝑈t 𝑈) = 𝑈)
118, 10syl 17 . . 3 ((𝐵 × 𝐵) = 𝑈 → (𝑈t 𝑈) = 𝑈)
121, 11eqtr2d 2860 . 2 ((𝐵 × 𝐵) = 𝑈𝑈 = (𝑈t (𝐵 × 𝐵)))
13 ussval.2 . . 3 𝑈 = (UnifSet‘𝑊)
143, 13ussval 22863 . 2 (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
1512, 14syl6eq 2875 1 ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3480  ∪ cuni 4825   × cxp 5541  ‘cfv 6344  (class class class)co 7146  Basecbs 16481  UnifSetcunif 16573   ↾t crest 16692  UnifStcuss 22857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-rest 16694  df-uss 22860 This theorem is referenced by:  tususs  22874  cnflduss  23958
 Copyright terms: Public domain W3C validator