P. 42 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4101-4200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	breq2d 4101	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12d 4102	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breq123d 4103	Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))
Theorem	breqdi 4104	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶𝐴𝐷)    ⇒   ⊢ (𝜑 → 𝐶𝐵𝐷)
Theorem	breqan12d 4105	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	breqan12rd 4106	Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
Theorem	eqnbrtrd 4107	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)
Theorem	nbrne1 4108	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶)
Theorem	nbrne2 4109	Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
⊢ ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴 ≠ 𝐵)
Theorem	eqbrtri 4110	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	eqbrtrd 4111	Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrri 4112	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴𝑅𝐶    ⇒   ⊢ 𝐵𝑅𝐶
Theorem	eqbrtrrd 4113	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐶)
Theorem	breqtri 4114	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrd 4115	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrri 4116	Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴𝑅𝐶
Theorem	breqtrrd 4117	Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	3brtr3i 4118	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr4i 4119	Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶𝑅𝐷
Theorem	3brtr3d 4120	Substitution of equality into both sides of a binary relation. (Contributed by NM, 18-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr4d 4121	Substitution of equality into both sides of a binary relation. (Contributed by NM, 21-Feb-2005.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr3g 4122	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	3brtr4g 4123	Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶𝑅𝐷)
Theorem	eqbrtrid 4124	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrid 4125	B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrid 4126	B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrid 4127	B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ 𝐴𝑅𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrdi 4128	A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqbrtrrdi 4129	A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵𝑅𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrdi 4130	A chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	breqtrrdi 4131	A chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	ssbrd 4132	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbr 4133	Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbri 4134	Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)
Theorem	nfbrd 4135	Deduction version of bound-variable hypothesis builder nfbr 4136. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝑅)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Theorem	nfbr 4136	Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴𝑅𝐵
Theorem	brab1 4137*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
Theorem	br0 4138	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
⊢ ¬ 𝐴∅𝐵
Theorem	brne0 4139	If two sets are in a binary relation, the relation cannot be empty. In fact, the relation is also inhabited, as seen at brm 4140. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)
Theorem	brm 4140*	If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
⊢ (𝐴𝑅𝐵 → ∃𝑥 𝑥 ∈ 𝑅)
Theorem	brun 4141	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))
Theorem	brin 4142	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))
Theorem	brdif 4143	The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
Theorem	sbcbrg 4144	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))
Theorem	sbcbr12g 4145*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶))
Theorem	sbcbr1g 4146*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶))
Theorem	sbcbr2g 4147*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶))
Theorem	brralrspcev 4148*	Restricted existential specialization with a restricted universal quantifier over a relation, closed form. (Contributed by AV, 20-Aug-2022.)
⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 𝐴𝑅𝐵) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴𝑅𝑥)
Theorem	brimralrspcev 4149*	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.)
⊢ ((𝐵 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝐵) → 𝜓)) → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝜑 ∧ 𝐴𝑅𝑥) → 𝜓))
2.1.23  Ordered-pair class abstractions (class builders)
Syntax	copab 4150	Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Syntax	cmpt 4151	Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥 ∈ 𝐴 ↦ 𝐵)
Definition	df-opab 4152*	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 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Definition	df-mpt 4153*	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.)
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
Theorem	opabss 4154*	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
Theorem	opabbid 4155	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Theorem	opabbidv 4156*	Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Theorem	opabbii 4157	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Theorem	nfopab 4158*	Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	nfopab1 4159	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.)
⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	nfopab2 4160	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.)
⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	cbvopab 4161*	Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Theorem	cbvopabv 4162*	Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Theorem	cbvopab1 4163*	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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Theorem	cbvopab2 4164*	Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
Theorem	cbvopab1s 4165*	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}
Theorem	cbvopab1v 4166*	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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Theorem	cbvopab2v 4167*	Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑧⟩ ∣ 𝜓}
Theorem	csbopabg 4168*	Move substitution into a class abstraction. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
Theorem	unopab 4169	Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}
Theorem	mpteq12f 4170	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12dva 4171*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12dv 4172*	An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12 4173*	An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq1 4174*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	mpteq1d 4175*	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	mpteq2ia 4176	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
Theorem	mpteq2i 4177	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
Theorem	mpteq12i 4178	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    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)
Theorem	mpteq2da 4179	Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	mpteq2dva 4180*	Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	mpteq2dv 4181*	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
Theorem	nfmpt 4182*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)
Theorem	nfmpt1 4183	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	cbvmptf 4184*	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.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmpt 4185*	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.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptv 4186*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	mptv 4187*	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
Syntax	wtr 4188	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 4189	Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 4190 (which is suggestive of the word "transitive"), dftr3 4192, dftr4 4193, and dftr5 4191. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
Theorem	dftr2 4190*	An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
Theorem	dftr5 4191*	An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 4192*	An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
Theorem	dftr4 4193	An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
⊢ (Tr 𝐴 ↔ 𝐴 ⊆ 𝒫 𝐴)
Theorem	treq 4194	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 4195	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 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
Theorem	trel3 4196	In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴))
Theorem	trss 4197	An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	trin 4198	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 4199	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 4200	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V