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Theorem tusval 24289
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 24282 . 2 toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
2 simpr 484 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
32unieqd 4924 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
43dmeqd 5918 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = dom 𝑈)
54opeq2d 4884 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), dom 𝑢⟩ = ⟨(Base‘ndx), dom 𝑈⟩)
62opeq2d 4884 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(UnifSet‘ndx), 𝑢⟩ = ⟨(UnifSet‘ndx), 𝑈⟩)
75, 6preq12d 4745 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩})
82fveq2d 6910 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈))
98opeq2d 4884 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩ = ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)
107, 9oveq12d 7448 . 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)⟩))
11 elfvunirn 6938 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
12 ovexd 7465 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩) ∈ V)
131, 10, 11, 12fvmptd2 7023 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 395   = wceq 1536  wcel 2105  Vcvv 3477  {cpr 4632  cop 4636   cuni 4911  dom cdm 5688  ran crn 5689  cfv 6562  (class class class)co 7430   sSet csts 17196  ndxcnx 17226  Basecbs 17244  TopSetcts 17303  UnifSetcunif 17307  UnifOncust 24223  unifTopcutop 24254  toUnifSpctus 24279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-tus 24282
This theorem is referenced by:  tuslem  24290  tuslemOLD  24291
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