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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdifxp1 5801 Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶))

Theoremdifxp2 5802 Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))

Theoremdjudisj 5803* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((𝐴𝐵) = ∅ → ( 𝑥𝐴 ({𝑥} × 𝐶) ∩ 𝑦𝐵 ({𝑦} × 𝐷)) = ∅)

Theoremxpdifid 5804* The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.)
𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )

Theoremresdisj 5805 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)

Theoremrnxp 5806 The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
(𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)

Theoremdmxpss 5807 The domain of a Cartesian product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
dom (𝐴 × 𝐵) ⊆ 𝐴

Theoremrnxpss 5808 The range of a Cartesian product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ran (𝐴 × 𝐵) ⊆ 𝐵

Theoremrnxpid 5809 The range of a square Cartesian product. (Contributed by FL, 17-May-2010.)
ran (𝐴 × 𝐴) = 𝐴

Theoremssxpb 5810 A Cartesian product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.)
((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))

Theoremxp11 5811 The Cartesian product of nonempty classes is one-to-one. (Contributed by NM, 31-May-2008.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremxpcan 5812 Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
(𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Theoremxpcan2 5813 Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
(𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Theoremssrnres 5814 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Theoremrninxp 5815* Range of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)

Theoremdminxp 5816* Domain of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.)
(dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)

Theoremimainrect 5817 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)
((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)

Theoremxpima 5818 The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
((𝐴 × 𝐵) “ 𝐶) = if((𝐴𝐶) = ∅, ∅, 𝐵)

Theoremxpima1 5819 The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Theoremxpima2 5820 The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝐴𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Theoremxpimasn 5821 The image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theoremsossfld 5822 The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ∅ Or {𝐵}). (Contributed by Mario Carneiro, 27-Apr-2015.)
((𝑅 Or 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))

Theoremsofld 5823 The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)
((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Theoremcnvcnv3 5824* The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}

Theoremdfrel2 5825 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
(Rel 𝑅𝑅 = 𝑅)

Theoremdfrel4v 5826* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6489 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
(Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Theoremdfrel4 5827* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6489 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
𝑥𝑅    &   𝑦𝑅       (Rel 𝑅𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})

Theoremcnvcnv 5828 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.)
𝐴 = (𝐴 ∩ (V × V))

Theoremcnvcnv2 5829 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
𝐴 = (𝐴 ↾ V)

Theoremcnvcnvss 5830 The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
𝐴𝐴

Theoremcnveqb 5831 Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵𝐴 = 𝐵))

Theoremcnveq0 5832 A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
(Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))

Theoremdfrel3 5833 Alternate definition of relation. (Contributed by NM, 14-May-2008.)
(Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Theoremelid 5834* Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 5833 after elrid 5695. (Contributed by BJ, 28-Aug-2022.)
(𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)

Theoremdmresv 5835 The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
dom (𝐴 ↾ V) = dom 𝐴

Theoremrnresv 5836 The range of a universal restriction. (Contributed by NM, 14-May-2008.)
ran (𝐴 ↾ V) = ran 𝐴

Theoremdfrn4 5837 Range defined in terms of image. (Contributed by NM, 14-May-2008.)
ran 𝐴 = (𝐴 “ V)

Theoremcsbrn 5838 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵

Theoremrescnvcnv 5839 The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcnvcnvres 5840 The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremimacnvcnv 5841 The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremdmsnn0 5842 The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)

Theoremrnsnn0 5843 The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.)
(𝐴 ∈ (V × V) ↔ ran {𝐴} ≠ ∅)

Theoremdmsn0 5844 The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
dom {∅} = ∅

Theoremcnvsn0 5845 The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
{∅} = ∅

Theoremdmsn0el 5846 The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
(∅ ∈ 𝐴 → dom {𝐴} = ∅)

Theoremrelsn2 5847 A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022.)
(𝐴𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))

Theoremdmsnopg 5848 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐵𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})

Theoremdmsnopss 5849 The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.)
dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Theoremdmpropg 5850 The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐵𝑉𝐷𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Theoremdmsnop 5851 The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐵 ∈ V       dom {⟨𝐴, 𝐵⟩} = {𝐴}

Theoremdmprop 5852 The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Theoremdmtpop 5853 The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐷 ∈ V    &   𝐹 ∈ V       dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Theoremcnvcnvsn 5854 Double converse of a singleton of an ordered pair. (Unlike cnvsn 5861, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Theoremdmsnsnsn 5855 The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
dom {{{𝐴}}} = {𝐴}

Theoremrnsnopg 5856 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, 30-Apr-2015.)
(𝐴𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})

Theoremrnpropg 5857 The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Theoremcnvsng 5858 Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) (Proof shortened by BJ, 12-Feb-2022.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Theoremrnsnop 5859 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 5860 Extract the first member of an ordered pair. (See op2nda 5863 to extract the second member, op1stb 5161 for an alternate version, and op1st 7437 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        dom {⟨𝐴, 𝐵⟩} = 𝐴

Theoremcnvsn 5861 Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by BJ, 12-Feb-2022.)
𝐴 ∈ V    &   𝐵 ∈ V       {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}

Theoremop2ndb 5862 Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5161 to extract the first member, op2nda 5863 for an alternate version, and op2nd 7438 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        {⟨𝐴, 𝐵⟩} = 𝐵

Theoremop2nda 5863 Extract the second member of an ordered pair. (See op1sta 5860 to extract the first member, op2ndb 5862 for an alternate version, and op2nd 7438 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V        ran {⟨𝐴, 𝐵⟩} = 𝐵

Theoremopswap 5864 Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)
{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴

Theoremcnvresima 5865 An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
((𝐹𝐴) “ 𝐵) = ((𝐹𝐵) ∩ 𝐴)

Theoremresdm2 5866 A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom 𝐴) = 𝐴

Theoremresdmres 5867 Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremresresdm 5868 A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
(𝐹 = (𝐸𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))

Theoremimadmres 5869 The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
(𝐴 “ dom (𝐴𝐵)) = (𝐴𝐵)

Theoremmptpreima 5870* The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐹 = (𝑥𝐴𝐵)       (𝐹𝐶) = {𝑥𝐴𝐵𝐶}

Theoremmptiniseg 5871* 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 5872 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 5873* The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
𝐹 = (𝑥𝐴𝐵)       dom 𝐹𝐴

Theoremdmmptg 5874* 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 5875 A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
Rel (𝐴𝐵)

Theoremdfco2 5876* 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 5877* Generalization of dfco2 5876, 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 5878 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremcoundir 5879 Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremcores 5880 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 5881 Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵𝐶))

Theoremimaco 5882 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
((𝐴𝐵) “ 𝐶) = (𝐴 “ (𝐵𝐶))

Theoremrnco 5883 The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)

Theoremrnco2 5884 The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
ran (𝐴𝐵) = (𝐴 “ ran 𝐵)

Theoremdmco 5885 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Theoremcoeq0 5886 A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 5878 and coundir 5879 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Theoremcoiun 5887* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremcocnvcnv1 5888 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcocnvcnv2 5889 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcores2 5890 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 5891 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅

Theoremco01 5892 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅

Theoremcoi1 5893 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 5894 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 5895 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 5896 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 5897 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))

Theoremrelssdmrn 5898 A relation is included in the Cartesian 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 5899 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 5900 Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)

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