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Theorem tusval 22051
Description: The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
tusval (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))

Proof of Theorem tusval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 df-tus 22043 . . 3 toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
21a1i 11 . 2 (𝑈 ∈ (UnifOn‘𝑋) → toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩)))
3 simpr 477 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
43unieqd 4437 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
54dmeqd 5315 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = dom 𝑈)
65opeq2d 4400 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), dom 𝑢⟩ = ⟨(Base‘ndx), dom 𝑈⟩)
73opeq2d 4400 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(UnifSet‘ndx), 𝑢⟩ = ⟨(UnifSet‘ndx), 𝑈⟩)
86, 7preq12d 4267 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩})
93fveq2d 6182 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈))
109opeq2d 4400 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩ = ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)
118, 10oveq12d 6653 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
12 elrnust 22009 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
13 ovexd 6665 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩) ∈ V)
142, 11, 12, 13fvmptd 6275 1 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  {cpr 4170  cop 4174   cuni 4427  cmpt 4720  dom cdm 5104  ran crn 5105  cfv 5876  (class class class)co 6635  ndxcnx 15835   sSet csts 15836  Basecbs 15838  TopSetcts 15928  UnifSetcunif 15932  UnifOncust 21984  unifTopcutop 22015  toUnifSpctus 22040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-ov 6638  df-ust 21985  df-tus 22043
This theorem is referenced by:  tuslem  22052
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