Theorem tusval 23417

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.

Step	Hyp	Ref	Expression
1		df-tus 23410	. 2 ⊢ toUnifSp = (𝑢 ∈ ∪ ran UnifOn ↦ ({⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
2		simpr 485	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
3	2	unieqd 4853	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈)
4	3	dmeqd 5814	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈)
5	4	opeq2d 4811	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), dom ∪ 𝑢⟩ = ⟨(Base‘ndx), dom ∪ 𝑈⟩)
6	2	opeq2d 4811	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(UnifSet‘ndx), 𝑢⟩ = ⟨(UnifSet‘ndx), 𝑈⟩)
7	5, 6	preq12d 4677	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), dom ∪ 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩})
8	2	fveq2d 6778	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (unifTop‘𝑢) = (unifTop‘𝑈))
9	8	opeq2d 4811	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩ = ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)
10	7, 9	oveq12d 7293	. 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		elrnust 23376	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
12		ovexd 7310	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩) ∈ V)
13	1, 10, 11, 12	fvmptd2 6883	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {cpr 4563  ⟨cop 4567  ∪ cuni 4839  dom cdm 5589  ran crn 5590  ‘cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  Basecbs 16912  TopSetcts 16968  UnifSetcunif 16972  UnifOncust 23351  unifTopcutop 23382  toUnifSpctus 23407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278  df-ust 23352  df-tus 23410

This theorem is referenced by:  tuslem  23418  tuslemOLD  23419