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Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnsnop 4801 The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ran {⟨𝐴, 𝐵⟩} = {𝐵}

Theoremop1sta 4802 Extract the first member of an ordered pair. (See op2nda 4805 to extract the second member and op1stb 4209 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        dom {⟨𝐴, 𝐵⟩} = 𝐴

Theoremcnvsn 4803 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Theoremop2ndb 4804 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4209 to extract the first member and op2nda 4805 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        {⟨𝐴, 𝐵⟩} = 𝐵

Theoremop2nda 4805 Extract the second member of an ordered pair. (See op1sta 4802 to extract the first member and op2ndb 4804 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V        ran {⟨𝐴, 𝐵⟩} = 𝐵

Theoremcnvsng 4806 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Theoremopswapg 4807 Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)

Theoremelxp4 4808 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4809. (Contributed by NM, 17-Feb-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))

Theoremelxp5 4809 Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4808 when the double intersection does not create class existence problems (caused by int0 3629). (Contributed by NM, 1-Aug-2004.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ 𝐴, ran {𝐴}⟩ ∧ ( 𝐴𝐵 ran {𝐴} ∈ 𝐶)))

Theoremcnvresima 4810 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)

Theoremresdm2 4811 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom 𝐴) = 𝐴

Theoremresdmres 4812 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremimadmres 4813 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremmptpreima 4814* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐹𝐶) = {𝑥𝐴𝐵𝐶}

Theoremmptiniseg 4815* Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐶𝑉 → (𝐹 “ {𝐶}) = {𝑥𝐴𝐵 = 𝐶})

Theoremdmmpt 4816 The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹 = {𝑥𝐴𝐵 ∈ V}

Theoremdmmptss 4817* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴

Theoremdmmptg 4818* The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
(∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)

Theoremrelco 4819 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (𝐴𝐵)

Theoremdfco2 4820* Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
(𝐴𝐵) = 𝑥 ∈ V ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Theoremdfco2a 4821* Generalization of dfco2 4820, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴𝐵) = 𝑥𝐶 ((𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Theoremcoundi 4822 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremcoundir 4823 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremcores 4824 Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran 𝐵𝐶 → ((𝐴𝐶) ∘ 𝐵) = (𝐴𝐵))

Theoremresco 4825 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Theoremimaco 4826 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))

Theoremrnco 4827 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)

Theoremrnco2 4828 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (𝐴𝐵) = (𝐴 “ ran 𝐵)

Theoremdmco 4829 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Theoremcoiun 4830* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremcocnvcnv1 4831 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcocnvcnv2 4832 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcores2 4833 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))

Theoremco02 4834 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅

Theoremco01 4835 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅

Theoremcoi1 4836 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)

Theoremcoi2 4837 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Theoremcoires1 4838 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)

Theoremcoass 4839 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Theoremrelcnvtr 4840 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Theoremrelssdmrn 4841 A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))

Theoremcnvssrndm 4842 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐴 ⊆ (ran 𝐴 × dom 𝐴)

Theoremcossxp 4843 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)

Theoremrelrelss 4844 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Theoremunielrel 4845 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)

Theoremrelfld 4846 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Theoremrelresfld 4847 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)

Theoremrelcoi2 4848 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)

Theoremrelcoi1 4849 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)

Theoremunidmrn 4850 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)

Theoremrelcnvfld 4851 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)

Theoremdfdm2 4852 Alternate definition of domain df-dm 4355 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)

Theoremunixpm 4853* The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (𝐴 × 𝐵) = (𝐴𝐵))

Theoremunixp0im 4854 The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)

Theoremcnvexg 4855 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
(𝐴𝑉𝐴 ∈ V)

Theoremcnvex 4856 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
𝐴 ∈ V       𝐴 ∈ V

Theoremrelcnvexb 4857 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
(Rel 𝑅 → (𝑅 ∈ V ↔ 𝑅 ∈ V))

Theoremressn 4858 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))

Theoremcnviinm 4859* The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
(∃𝑦 𝑦𝐴 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)

Theoremcnvpom 4860* The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐴))

Theoremcnvsom 4861* The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
(∃𝑥 𝑥𝐴 → (𝑅 Or 𝐴𝑅 Or 𝐴))

Theoremcoexg 4862 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoremcoex 4863 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V

Theoremxpcom 4864* Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
(∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

2.6.7  Definite description binder (inverted iota)

Syntaxcio 4865 Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)

Theoremiotajust 4866* Soundness justification theorem for df-iota 4867. (Contributed by Andrew Salmon, 29-Jun-2011.)
{𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}

Definitiondf-iota 4867* Define Russell's definition description binder, which can be read as "the unique 𝑥 such that 𝜑," where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see iotaval 4878); otherwise, it evaluates to the empty set (see iotanul 4882). Russell used the inverted iota symbol to represent the binder.

Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4890 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF.

(Contributed by Andrew Salmon, 30-Jun-2011.)

(℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}

Theoremdfiota2 4868* Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
(℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}

Theoremnfiota1 4869 Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥(℩𝑥𝜑)

Theoremnfiotadxy 4870* Deduction version of nfiotaxy 4871. (Contributed by Jim Kingdon, 21-Dec-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))

Theoremnfiotaxy 4871* Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.)
𝑥𝜑       𝑥(℩𝑦𝜑)

Theoremcbviota 4872 Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremcbviotav 4873* Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)

Theoremsb8iota 4874 Variable substitution in description binder. Compare sb8eu 1913. (Contributed by NM, 18-Mar-2013.)
𝑦𝜑       (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Theoremiotaeq 4875 Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑))

Theoremiotabi 4876 Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
(∀𝑥(𝜑𝜓) → (℩𝑥𝜑) = (℩𝑥𝜓))

Theoremuniabio 4877* Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
(∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} = 𝑦)

Theoremiotaval 4878* Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
(∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Theoremiotauni 4879 Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiotaint 4880 Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})

Theoremiota1 4881 Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Theoremiotanul 4882 Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)
(¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)

Theoremeuiotaex 4883 Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Theoremiotass 4884* Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
(∀𝑥(𝜑𝑥𝐴) → (℩𝑥𝜑) ⊆ 𝐴)

Theoremiota4 4885 Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)

Theoremiota4an 4886 Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
(∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)

Theoremiota5 4887* A method for computing iota. (Contributed by NM, 17-Sep-2013.)
((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremiotabidv 4888* Formula-building deduction rule for iota. (Contributed by NM, 20-Aug-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Theoremiotabii 4889 Formula-building deduction rule for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝜑𝜓)       (℩𝑥𝜑) = (℩𝑥𝜓)

Theoremiotacl 4890 Membership law for descriptions.

This can useful for expanding an unbounded iota-based definition (see df-iota 4867).

(Contributed by Andrew Salmon, 1-Aug-2011.)

(∃!𝑥𝜑 → (℩𝑥𝜑) ∈ {𝑥𝜑})

Theoremiota2df 4891 A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑𝑥𝐵)       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2d 4892* A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)
(𝜑𝐵𝑉)    &   (𝜑 → ∃!𝑥𝜓)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Theoremiota2 4893* The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Theoremsniota 4894 A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
(∃!𝑥𝜑 → {𝑥𝜑} = {(℩𝑥𝜑)})

Theoremcsbiotag 4895* Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
(𝐴𝑉𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))

2.6.8  Functions

Syntaxwfun 4896 Extend the definition of a wff to include the function predicate. (Read: 𝐴 is a function.)
wff Fun 𝐴

Syntaxwfn 4897 Extend the definition of a wff to include the function predicate with a domain. (Read: 𝐴 is a function on 𝐵.)
wff 𝐴 Fn 𝐵

Syntaxwf 4898 Extend the definition of a wff to include the function predicate with domain and codomain. (Read: 𝐹 maps 𝐴 into 𝐵.)
wff 𝐹:𝐴𝐵

Syntaxwf1 4899 Extend the definition of a wff to include one-to-one functions. (Read: 𝐹 maps 𝐴 one-to-one into 𝐵.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
wff 𝐹:𝐴1-1𝐵

Syntaxwfo 4900 Extend the definition of a wff to include onto functions. (Read: 𝐹 maps 𝐴 onto 𝐵.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
wff 𝐹:𝐴onto𝐵

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