P. 373 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rnresequniqs 37201	The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.)
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅))
Theorem	n0el2 37202	Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.)
⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴)
Theorem	cnvepresex 37203	Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.)
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V)
Theorem	eccnvepex 37204	The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ [𝐴]◡ E ∈ V
Theorem	cnvepimaex 37205	The image of converse epsilon exists, proof via imaexALTV 37199 (see also cnvepima 37206 and uniexg 7730 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V)
Theorem	cnvepima 37206	The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴)
Theorem	inex3 37207	Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.)
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V)
Theorem	inxpex 37208	Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)
⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
Theorem	eqres 37209	Converting a class constant definition by restriction (like df-ers 37533 or df-parts 37635) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
⊢ 𝑅 = (𝑆 ↾ 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵)))
Theorem	brrabga 37210*	The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩𝑅𝐶 ↔ 𝜓))
Theorem	brcnvrabga 37211*	The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ 𝜑}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅⟨𝐵, 𝐶⟩ ↔ 𝜓))
Theorem	opideq 37212	Equality conditions for ordered pairs ⟨𝐴, 𝐴⟩ and ⟨𝐵, 𝐵⟩. (Contributed by Peter Mazsa, 22-Jul-2019.) (Revised by Thierry Arnoux, 16-Feb-2022.)
⊢ (𝐴 ∈ 𝑉 → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))
Theorem	iss2 37213	A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴)))
Theorem	eldmcnv 37214*	Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	dfrel5 37215	Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
Theorem	dfrel6 37216	Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
Theorem	cnvresrn 37217	Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.)
⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅
Theorem	relssinxpdmrn 37218	Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023.)
⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))
Theorem	cnvref4 37219	Two ways to say that a relation is a subclass. (Contributed by Peter Mazsa, 11-Apr-2023.)
⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆))
Theorem	cnvref5 37220*	Two ways to say that a relation is a subclass of the identity relation. (Contributed by Peter Mazsa, 26-Jun-2019.)
⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦)))
Theorem	ecin0 37221*	Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥)))
Theorem	ecinn0 37222*	Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
Theorem	ineleq 37223*	Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦))
Theorem	inecmo 37224*	Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧))
Theorem	inecmo2 37225*	Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.) (Revised by Peter Mazsa, 2-Sep-2021.)
⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	ineccnvmo 37226*	Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
Theorem	alrmomorn 37227	Equivalence of an "at most one" and an "at most one" restricted to the range inside a universal quantification. (Contributed by Peter Mazsa, 3-Sep-2021.)
⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦)
Theorem	alrmomodm 37228*	Equivalence of an "at most one" and an "at most one" restricted to the domain inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥))
Theorem	ineccnvmo2 37229*	Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021.)
⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥)
Theorem	inecmo3 37230*	Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
Theorem	moeu2 37231	Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)
⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
Theorem	mopickr 37232	"At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2633) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)
⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
Theorem	moantr 37233	Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.)
⊢ (∃*𝑥𝜓 → ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜓 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜒)))
Theorem	brabidgaw 37234*	The law of concretion for a binary relation. Special case of brabga 5535. Version of brabidga 37235 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝑥𝑅𝑦 ↔ 𝜑)
Theorem	brabidga 37235	The law of concretion for a binary relation. Special case of brabga 5535. Usage of this theorem is discouraged because it depends on ax-13 2372, see brabidgaw 37234 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ (𝑥𝑅𝑦 ↔ 𝜑)
Theorem	inxp2 37236*	Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.)
⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)}
Theorem	opabf 37237	A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) (Proof shortened by Thierry Arnoux, 18-Feb-2022.)
⊢ ¬ 𝜑    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅
Theorem	ec0 37238	The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.)
⊢ [𝐴]∅ = ∅
Theorem	0qs 37239	Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.)
⊢ (∅ / 𝑅) = ∅
Theorem	brcnvin 37240	Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴)))
21.24.3  Range Cartesian product
Definition	df-xrn 37241	Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 37242 and brxrn 37244. This is Scott Fenton's df-txp 34826 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34826. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
Theorem	xrnss3v 37242	A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 34850 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34850. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
Theorem	xrnrel 37243	A range Cartesian product is a relation. This is Scott Fenton's txprel 34851 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34851. (Contributed by Scott Fenton, 31-Mar-2012.)
⊢ Rel (𝐴 ⋉ 𝐵)
Theorem	brxrn 37244	Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 37242, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶)))
Theorem	brxrn2 37245*	A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
Theorem	dfxrn2 37246*	Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.)
⊢ (𝑅 ⋉ 𝑆) = ◡{⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
Theorem	xrneq1 37247	Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
Theorem	xrneq1i 37248	Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)
Theorem	xrneq1d 37249	Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶))
Theorem	xrneq2 37250	Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
Theorem	xrneq2i 37251	Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)
Theorem	xrneq2d 37252	Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵))
Theorem	xrneq12 37253	Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
Theorem	xrneq12i 37254	Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)
Theorem	xrneq12d 37255	Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷))
Theorem	elecxrn 37256*	Elementhood in the (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
Theorem	ecxrn 37257*	The (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
Theorem	disjressuc2 37258*	Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢 ∈ 𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
Theorem	disjecxrn 37259	Two ways of saying that (𝑅 ⋉ 𝑆)-cosets are disjoint. (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, 21-Aug-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅)))
Theorem	disjecxrncnvep 37260	Two ways of saying that cosets are disjoint, special case of disjecxrn 37259. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 25-Aug-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝐴 ∩ 𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))
Theorem	disjsuc2 37261*	Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Theorem	xrninxp 37262*	Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.)
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))}
Theorem	xrninxp2 37263*	Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.)
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
Theorem	xrninxpex 37264	Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
Theorem	inxpxrn 37265	Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.)
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
Theorem	br1cnvxrn2 37266*	The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ (𝐵 ∈ 𝑉 → (𝐴◡(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)))
Theorem	elec1cnvxrn2 37267*	Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.)
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧)))
Theorem	rnxrn 37268*	Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.)
⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
Theorem	rnxrnres 37269*	Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.)
⊢ ran (𝑅 ⋉ (𝑆 ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)}
Theorem	rnxrncnvepres 37270*	Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)}
Theorem	rnxrnidres 37271*	Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.)
⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)}
Theorem	xrnres 37272	Two ways to express restriction of range Cartesian product, see also xrnres2 37273, xrnres3 37274. (Contributed by Peter Mazsa, 5-Jun-2021.)
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆)
Theorem	xrnres2 37273	Two ways to express restriction of range Cartesian product, see also xrnres 37272, xrnres3 37274. (Contributed by Peter Mazsa, 6-Sep-2021.)
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
Theorem	xrnres3 37274	Two ways to express restriction of range Cartesian product, see also xrnres 37272, xrnres2 37273. (Contributed by Peter Mazsa, 28-Mar-2020.)
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴))
Theorem	xrnres4 37275	Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.)
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴))))
Theorem	xrnresex 37276	Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V)
Theorem	xrnidresex 37277	Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V)
Theorem	xrncnvepresex 37278	Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V)
Theorem	brin2 37279	Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩))
Theorem	brin3 37280	Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}}))
21.24.4  Cosets by ` R `
Definition	df-coss 37281*	Define the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑢𝑅𝑥 and 𝑢𝑅𝑦 hold, i.e., both 𝑥 and 𝑦 are are elements of the 𝑅 -coset of 𝑢 (see dfcoss2 37283 and the comment of dfec2 8706). 𝑅 is usually a relation. This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to 𝑅, the right side relate to ≀ 𝑅 (see e.g. pet 37721). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 37284) or to the range of a range Cartesian product of classes (cf. dfcoss4 37285), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 37283. Technically, we can define it via composition (dfcoss3 37284) or as the range of a range Cartesian product (dfcoss4 37285), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 37551, df-funALTV 37552) and disjoints (dfdisjs 37578, dfdisjs2 37579, df-disjALTV 37575, dfdisjALTV2 37584) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Definition	df-coels 37282	Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 37300. Possible definitions are the special cases of dfcoss3 37284 and dfcoss4 37285. (Contributed by Peter Mazsa, 20-Nov-2019.)
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
Theorem	dfcoss2 37283*	Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8706). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}
Theorem	dfcoss3 37284	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37281). (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)
Theorem	dfcoss4 37285	Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37281). (Contributed by Peter Mazsa, 12-Jul-2021.)
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)
Theorem	cosscnv 37286*	Class of cosets by the converse of 𝑅 (Contributed by Peter Mazsa, 17-Jun-2020.)
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
Theorem	coss1cnvres 37287*	Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}
Theorem	coss2cnvepres 37288*	Special case of coss1cnvres 37287. (Contributed by Peter Mazsa, 8-Jun-2020.)
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))}
Theorem	cossex 37289	If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V)
Theorem	cosscnvex 37290	If 𝐴 is a set then the class of cosets by the converse of 𝐴 is a set. (Contributed by Peter Mazsa, 18-Oct-2019.)
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V)
Theorem	1cosscnvepresex 37291	Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V)
Theorem	1cossxrncnvepresex 37292	Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V)
Theorem	relcoss 37293	Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ Rel ≀ 𝑅
Theorem	relcoels 37294	Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
⊢ Rel ∼ 𝐴
Theorem	cossss 37295	Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Theorem	cosseq 37296	Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)
Theorem	cosseqi 37297	Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ≀ 𝐴 = ≀ 𝐵
Theorem	cosseqd 37298	Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵)
Theorem	1cossres 37299*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 37300*	Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)}