P. 52 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ssbrd 5101	Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbr 5102	Implication from a subclass relationship of binary relations. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
Theorem	ssbri 5103	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 5104	Deduction version of bound-variable hypothesis builder nfbr 5105. (Contributed by NM, 13-Dec-2005.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝑅)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴𝑅𝐵)
Theorem	nfbr 5105	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 5106*	Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
Theorem	br0 5107	The empty binary relation never holds. (Contributed by NM, 23-Aug-2018.)
⊢ ¬ 𝐴∅𝐵
Theorem	brne0 5108	If two sets are in a binary relation, the relation cannot be empty. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ (𝐴𝑅𝐵 → 𝑅 ≠ ∅)
Theorem	brun 5109	The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))
Theorem	brin 5110	The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))
Theorem	brdif 5111	The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵))
Theorem	sbcbr123 5112	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Revised by NM, 22-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)
Theorem	sbcbr 5113*	Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵⦋𝐴 / 𝑥⦌𝑅𝐶)
Theorem	sbcbr12g 5114*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅⦋𝐴 / 𝑥⦌𝐶))
Theorem	sbcbr1g 5115*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵𝑅𝐶))
Theorem	sbcbr2g 5116*	Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ 𝐵𝑅⦋𝐴 / 𝑥⦌𝐶))
Theorem	brsymdif 5117	Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
Theorem	brralrspcev 5118*	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 5119*	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))
2.1.24  Ordered-pair class abstractions (class builders)
Syntax	copab 5120	Extend class notation to include ordered-pair class abstraction (class builder).
class {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Definition	df-opab 5121*	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 does not require it (see dfid2 5457 for a case where they are not distinct). The brace notation is called "class abstraction" by Quine; it is also called "class builder" in the literature. An alternate definition using no existential quantifiers is shown by dfopab2 7744. An example is given by ex-opab 28205. (Contributed by NM, 4-Jul-1994.)
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
Theorem	opabss 5122*	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 5123	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 5124*	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 5125	Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓}
Theorem	nfopab 5126*	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 5127	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 5128	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 5129*	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 5130*	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 5131*	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.) Add disjoint variable condition to avoid ax-13 2386. See cbvopab1g 5132 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Theorem	cbvopab1g 5132*	Change first bound variable in an ordered-pair class abstraction, using explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvopab1 5131 for a version with more disjoint variable conditions, but not requiring ax-13 2386. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Theorem	cbvopab2 5133*	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 5134*	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 5135*	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 5136*	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	unopab 5137	Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ 𝜓)}
2.1.25  Functions in maps-to notation
Syntax	cmpt 5138	Extend the definition of a class to include maps-to notation for defining a function via a rule.
class (𝑥 ∈ 𝐴 ↦ 𝐵)
Definition	df-mpt 5139*	Define maps-to notation for defining a function via a rule. Read as "the function which maps 𝑥 (in 𝐴) to 𝐵(𝑥)". The class expression 𝐵 is the value of the function at 𝑥 and normally contains the variable 𝑥. An example is the square function for complex numbers, (𝑥 ∈ ℂ ↦ (𝑥↑2)). 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	mpteq12df 5140	An equality inference for the maps-to notation. Compare mpteq12dv 5143. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12f 5141	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 5142*	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 5143*	An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Drop ax-10 2141 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12dvOLD 5144*	Obsolete version of mpteq12dv 5143 as of 1-Dec-2023. (Contributed by NM, 24-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
Theorem	mpteq12 5145*	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 5146*	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 5147*	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	mpteq1i 5148*	An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
Theorem	mpteq2ia 5149	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 5150	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 5151	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 5152	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 5153*	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 5154*	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 5155*	Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)
Theorem	nfmpt1 5156	Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	cbvmptf 5157*	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.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2386. See cbvmptfg 5158 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptfg 5158	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. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvmptf 5157 for a version with more disjoint variable conditions, but not requiring ax-13 2386. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmpt 5159*	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.) Add disjoint variable condition to avoid ax-13 2386. See cbvmptg 5160 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptg 5160*	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. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvmpt 5159 for a version with more disjoint variable conditions, but not requiring ax-13 2386. (Contributed by NM, 11-Sep-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptv 5161*	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2386. See cbvmptvg 5162 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	cbvmptvg 5162*	Rule to change the bound variable in a maps-to function, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvmptv 5161 for a version with more disjoint variable conditions, but not requiring ax-13 2386. (Contributed by Mario Carneiro, 19-Feb-2013.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
Theorem	mptv 5163*	Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
⊢ (𝑥 ∈ V ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐵}
2.1.26  Transitive classes
Syntax	wtr 5164	Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr 𝐴
Definition	df-tr 5165	Define the transitive class predicate. Not to be confused with a transitive relation (see cotr 5966). Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 5166 (which is suggestive of the word "transitive"), dftr3 5168, dftr4 5169, dftr5 5167, and (when 𝐴 is a set) unisuc 6261. 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 5166*	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 5167*	An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
Theorem	dftr3 5168*	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 5169	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 5170	Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Theorem	trel 5171	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 5172	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 5173	An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.)
⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	trin 5174	The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
Theorem	tr0 5175	The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
⊢ Tr ∅
Theorem	trv 5176	The universe is transitive. (Contributed by NM, 14-Sep-2003.)
⊢ Tr V
Theorem	triun 5177	An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	truni 5178*	The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)
Theorem	triin 5179	An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)
Theorem	trint 5180*	The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.)
⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)
Theorem	trintss 5181	Any nonempty transitive class includes its intersection. Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the nonemptiness hypothesis). (Contributed by Scott Fenton, 3-Mar-2011.) (Proof shortened by Andrew Salmon, 14-Nov-2011.)
⊢ ((Tr 𝐴 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ 𝐴)
2.2  ZF Set Theory - add the Axiom of Replacement
2.2.1  Introduce the Axiom of Replacement
Axiom	ax-rep 5182*	Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19. It tells us that the image of any set under a function is also a set (see the variant funimaex 6435). Although 𝜑 may be any wff whatsoever, this axiom is useful (i.e. its antecedent is satisfied) when we are given some function and 𝜑 encodes the predicate "the value of the function at 𝑤 is 𝑧". Thus, 𝜑 will ordinarily have free variables 𝑤 and 𝑧- think of it informally as 𝜑(𝑤, 𝑧). We prefix 𝜑 with the quantifier ∀𝑦 in order to "protect" the axiom from any 𝜑 containing 𝑦, thus allowing us to eliminate any restrictions on 𝜑. Another common variant is derived as axrep5 5188, where you can find some further remarks. A slightly more compact version is shown as axrep2 5185. A quite different variant is zfrep6 7650, which if used in place of ax-rep 5182 would also require that the Separation Scheme axsep 5194 be stated as a separate axiom. There is a very strong generalization of Replacement that doesn't demand function-like behavior of 𝜑. Two versions of this generalization are called the Collection Principle cp 9314 and the Boundedness Axiom bnd 9315. Many developments of set theory distinguish the uses of Replacement from uses of the weaker axioms of Separation axsep 5194, Null Set axnul 5201, and Pairing axpr 5320, all of which we derive from Replacement. In order to make it easier to identify the uses of those redundant axioms, we restate them as axioms ax-sep 5195, ax-nul 5202, and ax-pr 5321 below the theorems that prove them. (Contributed by NM, 23-Dec-1993.)
⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	axrep1 5183*	The version of the Axiom of Replacement used in the Metamath Solitaire applet https://us.metamath.org/mmsolitaire/mms.html. Equivalence is shown via the path ax-rep 5182 → axrep1 5183 → axrep2 5185 → axrepnd 10010 → zfcndrep 10030 = ax-rep 5182. (Contributed by NM, 19-Nov-2005.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-13 2386. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ 𝜑)))
Theorem	axreplem 5184*	Lemma for axrep2 5185 and axrep3 5186. (Contributed by BJ, 6-Aug-2022.)
⊢ (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))))
Theorem	axrep2 5185*	Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on 𝜑. (Contributed by NM, 15-Aug-2003.) Remove dependency on ax-13 2386. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
Theorem	axrep3 5186*	Axiom of Replacement slightly strengthened from axrep2 5185; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2386. (Revised by BJ, 31-May-2019.)
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
Theorem	axrep4 5187*	A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep5 5188*	Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us 𝜑 is analogous to a "function" from 𝑥 to 𝑦 (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set 𝑧 that corresponds to the "image" of 𝜑 restricted to some other set 𝑤. The hypothesis says 𝑧 must not be free in 𝜑. (Contributed by NM, 26-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑧𝜑    ⇒   ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
Theorem	axrep6 5189*	A condensed form of ax-rep 5182. (Contributed by SN, 18-Sep-2023.)
⊢ (∀𝑤∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
Theorem	zfrepclf 5190*	An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
⊢ Ⅎ𝑥𝐴    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))    ⇒   ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	zfrep3cl 5191*	An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))    ⇒   ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	zfrep4 5192*	A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.)
⊢ {𝑥 ∣ 𝜑} ∈ V    &   ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))    ⇒   ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V
2.2.2  Derive the Axiom of Separation
Theorem	axsepgfromrep 5193*	A more general version axsepg 5196 of the axiom scheme of separation ax-sep 5195 derived from the axiom scheme of replacement ax-rep 5182 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2120 to ax-13 2386. (Revised by SN, 25-Sep-2023.) Use ax-sep 5195 instead (or axsepg 5196 if the extra generality is needed). (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsep 5194*	Axiom scheme of separation ax-sep 5195 derived from the axiom scheme of replacement ax-rep 5182. The statement is identical to that of ax-sep 5195, and therefore shows that ax-sep 5195 is redundant when ax-rep 5182 is allowed. See ax-sep 5195 for more information. (Contributed by NM, 11-Sep-2006.) Use ax-sep 5195 instead. (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Axiom	ax-sep 5195*	Axiom scheme of separation. This is an axiom scheme of Zermelo and Zermelo-Fraenkel set theories. It was derived as axsep 5194 above and is therefore redundant in ZF set theory, which contains ax-rep 5182 as an axiom (contrary to Zermelo set theory). We state it as a separate axiom here so that some of its uses can be identified more easily. Some textbooks present the axiom scheme of separation as a separate axiom scheme in order to show that much of set theory can be derived without the stronger axiom scheme of replacement (which is not part of Zermelo set theory). The axiom scheme of separation is a weak form of Frege's axiom scheme of (unrestricted) comprehension, in that it conditions it with the condition 𝑥 ∈ 𝑧, so that it asserts the existence of a collection only if it is smaller than some other collection 𝑧 that already exists. This prevents Russell's paradox ru 3770. In some texts, this scheme is called "Aussonderung" (German for "separation") or "Subset Axiom". The variable 𝑥 can occur in the formula 𝜑, which in textbooks is often written 𝜑(𝑥). To specify this in the Metamath language, we omit the distinct variable condition ($d) that 𝑥 not occur in 𝜑. For a version using a class variable, see zfauscl 5197, which requires the axiom of extensionality as well as the axiom scheme of separation for its derivation. If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5254 shows (showing the necessity of that condition in zfauscl 5197). Scheme Sep of [BellMachover] p. 463. (Contributed by NM, 11-Sep-2006.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	axsepg 5196*	A more general version of the axiom scheme of separation ax-sep 5195, where variable 𝑧 can also occur (in addition to 𝑥) in formula 𝜑, which can therefore be thought of as 𝜑(𝑥, 𝑧). This version is derived from the more restrictive ax-sep 5195 with no additional set theory axioms. Note that it was also derived from ax-rep 5182 but without ax-sep 5195 as axsepgfromrep 5193. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) Remove dependency on ax-12 2173 and ax-13 2386 and shorten proof. (Revised by BJ, 6-Oct-2019.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	zfauscl 5197*	Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 5195, we invoke the Axiom of Extensionality (indirectly via vtocl 3559), which is needed for the justification of class variable notation. If we omit the requirement that 𝑦 not occur in 𝜑, we can derive a contradiction, as notzfaus 5254 shows. (Contributed by NM, 21-Jun-1993.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
Theorem	bm1.3ii 5198*	Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 5195. Similar to Theorem 1.3(ii) of [BellMachover] p. 463. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥∀𝑦(𝜑 → 𝑦 ∈ 𝑥)    ⇒   ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑)
Theorem	ax6vsep 5199*	Derive ax6v 1967 (a weakened version of ax-6 1966 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5195 and Extensionality ax-ext 2793. See ax6 2398 for the derivation of ax-6 1966 from ax6v 1967. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
2.2.3  Derive the Null Set Axiom
Theorem	axnulALT 5200*	Alternate proof of axnul 5201, proved from propositional calculus, ax-gen 1792, ax-4 1806, sp 2178, and ax-rep 5182. To check this, replace sp 2178 with the obsolete axiom ax-c5 36013 in the proof of axnulALT 5200 and type the Metamath program "MM> SHOW TRACE_BACK axnulALT / AXIOMS" command. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥