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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembreq2d 4001 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 4002 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreq123d 4003 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
 
Theorembreqdi 4004 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶𝐴𝐷)       (𝜑𝐶𝐵𝐷)
 
Theorembreqan12d 4005 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqan12rd 4006 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theoremeqnbrtrd 4007 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝑅𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremnbrne1 4008 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
 
Theoremnbrne2 4009 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
 
Theoremeqbrtri 4010 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐵𝑅𝐶       𝐴𝑅𝐶
 
Theoremeqbrtrd 4011 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrri 4012 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴 = 𝐵    &   𝐴𝑅𝐶       𝐵𝑅𝐶
 
Theoremeqbrtrrd 4013 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝑅𝐶)       (𝜑𝐵𝑅𝐶)
 
Theorembreqtri 4014 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐵 = 𝐶       𝐴𝑅𝐶
 
Theorembreqtrd 4015 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrri 4016 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
𝐴𝑅𝐵    &   𝐶 = 𝐵       𝐴𝑅𝐶
 
Theorembreqtrrd 4017 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theorem3brtr3i 4018 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝑅𝐷
 
Theorem3brtr4i 4019 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
𝐴𝑅𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝑅𝐷
 
Theorem3brtr3d 4020 Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4d 4021 Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
(𝜑𝐴𝑅𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr3g 4022 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝑅𝐷)
 
Theorem3brtr4g 4023 Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝑅𝐷)
 
Theoremeqbrtrid 4024 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴 = 𝐵    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrid 4025 B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrid 4026 B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
𝐴𝑅𝐵    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrid 4027 B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
𝐴𝑅𝐵    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrdi 4028 A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
(𝜑𝐴 = 𝐵)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theoremeqbrtrrdi 4029 A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
(𝜑𝐵 = 𝐴)    &   𝐵𝑅𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrdi 4030 A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
(𝜑𝐴𝑅𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝑅𝐶)
 
Theorembreqtrrdi 4031 A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
(𝜑𝐴𝑅𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝑅𝐶)
 
Theoremssbrd 4032 Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theoremssbri 4033 Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝐴𝐵       (𝐶𝐴𝐷𝐶𝐵𝐷)
 
Theoremnfbrd 4034 Deduction version of bound-variable hypothesis builder nfbr 4035. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝑅)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
 
Theoremnfbr 4035 Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝑅    &   𝑥𝐵       𝑥 𝐴𝑅𝐵
 
Theorembrab1 4036* Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
(𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
 
Theorembr0 4037 The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
¬ 𝐴𝐵
 
Theorembrne0 4038 If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4039. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
(𝐴𝑅𝐵𝑅 ≠ ∅)
 
Theorembrm 4039* If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
(𝐴𝑅𝐵 → ∃𝑥 𝑥𝑅)
 
Theorembrun 4040 The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrin 4041 The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembrdif 4042 The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
 
Theoremsbcbrg 4043 Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr12g 4044* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
 
Theoremsbcbr1g 4045* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
 
Theoremsbcbr2g 4046* Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
(𝐴𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))
 
Theorembrralrspcev 4047* Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 𝐴𝑅𝐵) → ∃𝑥𝑋𝑦𝑌 𝐴𝑅𝑥)
 
Theorembrimralrspcev 4048* Restricted existential specialization with a restricted universal quantifier over an implication with a relation in the antecedent, closed form. (Contributed by AV, 20-Aug-2022.)
((𝐵𝑋 ∧ ∀𝑦𝑌 ((𝜑𝐴𝑅𝐵) → 𝜓)) → ∃𝑥𝑋𝑦𝑌 ((𝜑𝐴𝑅𝑥) → 𝜓))
 
2.1.23  Ordered-pair class abstractions (class builders)
 
Syntaxcopab 4049 Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Syntaxcmpt 4050 Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥𝐴𝐵)
 
Definitiondf-opab 4051* Define the class abstraction of a collection of ordered pairs. Definition 3.3 of [Monk1] p. 34. Usually 𝑥 and 𝑦 are distinct, although the definition doesn't strictly require it. The brace notation is called "class abstraction" by Quine; it is also (more commonly) called a "class builder" in the literature. (Contributed by NM, 4-Jul-1994.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
 
Definitiondf-mpt 4052* Define maps-to notation for defining a function via a rule. Read as "the function defined by the map from 𝑥 (in 𝐴) to 𝐵(𝑥)". The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. Similar to the definition of mapping in [ChoquetDD] p. 2. (Contributed by NM, 17-Feb-2008.)
(𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
 
Theoremopabss 4053* The collection of ordered pairs in a class is a subclass of it. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
{⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
 
Theoremopabbid 4054 Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 
Theoremopabbidv 4055* Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 
Theoremopabbii 4056 Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
(𝜑𝜓)       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
 
Theoremnfopab 4057* Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
𝑧𝜑       𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremnfopab1 4058 The first abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremnfopab2 4059 The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
Theoremcbvopab 4060* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvopabv 4061* Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
Theoremcbvopab1 4062* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
Theoremcbvopab2 4063* Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
𝑧𝜑    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
 
Theoremcbvopab1s 4064* Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ [𝑧 / 𝑥]𝜑}
 
Theoremcbvopab1v 4065* Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
(𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
Theoremcbvopab2v 4066* Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
(𝑦 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
 
Theoremcsbopabg 4067* Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
 
Theoremunopab 4068 Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
 
Theoremmpteq12f 4069 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dva 4070* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(𝜑𝐴 = 𝐶)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12dv 4071* An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq12 4072* An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
Theoremmpteq1 4073* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝐴 = 𝐵 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq1d 4074* An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴𝐶) = (𝑥𝐵𝐶))
 
Theoremmpteq2ia 4075 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(𝑥𝐴𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq2i 4076 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
𝐵 = 𝐶       (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 
Theoremmpteq12i 4077 An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝐴 = 𝐶    &   𝐵 = 𝐷       (𝑥𝐴𝐵) = (𝑥𝐶𝐷)
 
Theoremmpteq2da 4078 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremmpteq2dva 4079* Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremmpteq2dv 4080* An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐵 = 𝐶)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
 
Theoremnfmpt 4081* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝑦𝐴𝐵)
 
Theoremnfmpt1 4082 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
𝑥(𝑥𝐴𝐵)
 
Theoremcbvmptf 4083* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmpt 4084* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremcbvmptv 4085* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
Theoremmptv 4086* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
 
2.1.24  Transitive classes
 
Syntaxwtr 4087 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
 
Definitiondf-tr 4088 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4089 (which is suggestive of the word "transitive"), dftr3 4091, dftr4 4092, and dftr5 4090. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 𝐴𝐴)
 
Theoremdftr2 4089* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
(Tr 𝐴 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝐴) → 𝑥𝐴))
 
Theoremdftr5 4090* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr 𝐴 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝐴)
 
Theoremdftr3 4091* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
 
Theoremdftr4 4092 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 
Theoremtreq 4093 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
 
Theoremtrel 4094 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐴) → 𝐵𝐴))
 
Theoremtrel3 4095 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr 𝐴 → ((𝐵𝐶𝐶𝐷𝐷𝐴) → 𝐵𝐴))
 
Theoremtrss 4096 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
(Tr 𝐴 → (𝐵𝐴𝐵𝐴))
 
Theoremtrin 4097 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴𝐵))
 
Theoremtr0 4098 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅
 
Theoremtrv 4099 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V
 
Theoremtriun 4100* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
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