P. 49 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4801-4900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnvcnv2 4801	The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
⊢ ◡◡𝐴 = (𝐴 ↾ V)
Theorem	cnvcnvss 4802	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
⊢ ◡◡𝐴 ⊆ 𝐴
Theorem	cnveqb 4803	Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))
Theorem	cnveq0 4804	A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))
Theorem	dfrel3 4805	Alternate definition of relation. (Contributed by NM, 14-May-2008.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
Theorem	dmresv 4806	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ dom (𝐴 ↾ V) = dom 𝐴
Theorem	rnresv 4807	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ ran (𝐴 ↾ V) = ran 𝐴
Theorem	dfrn4 4808	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
⊢ ran 𝐴 = (𝐴 “ V)
Theorem	csbrng 4809	Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)
Theorem	rescnvcnv 4810	The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
Theorem	cnvcnvres 4811	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
Theorem	imacnvcnv 4812	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)
Theorem	dmsnm 4813*	The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
Theorem	rnsnm 4814*	The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴})
Theorem	dmsn0 4815	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
⊢ dom {∅} = ∅
Theorem	cnvsn0 4816	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ◡{∅} = ∅
Theorem	dmsn0el 4817	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} = ∅)
Theorem	relsn2m 4818*	A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
Theorem	dmsnopg 4819	The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
Theorem	dmpropg 4820	The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
Theorem	dmsnop 4821	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}
Theorem	dmprop 4822	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}
Theorem	dmtpop 4823	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}
Theorem	cnvcnvsn 4824	Double converse of a singleton of an ordered pair. (Unlike cnvsn 4830, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}
Theorem	dmsnsnsng 4825	The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.)
⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴})
Theorem	rnsnopg 4826	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.)
⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
Theorem	rnpropg 4827	The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
Theorem	rnsnop 4828	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}
Theorem	op1sta 4829	Extract the first member of an ordered pair. (See op2nda 4832 to extract the second member and op1stb 4236 for an alternate version.) (Contributed by Raph Levien, 4-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
Theorem	cnvsn 4830	Converse of a singleton of an ordered pair. (Contributed by NM, 11-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Theorem	op2ndb 4831	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4236 to extract the first member and op2nda 4832 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵
Theorem	op2nda 4832	Extract the second member of an ordered pair. (See op1sta 4829 to extract the first member and op2ndb 4831 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵
Theorem	cnvsng 4833	Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Theorem	opswapg 4834	Swap the members of an ordered pair. (Contributed by Jim Kingdon, 16-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
Theorem	elxp4 4835	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 4836. (Contributed by NM, 17-Feb-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	elxp5 4836	Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4835 when the double intersection does not create class existence problems (caused by int0 3656). (Contributed by NM, 1-Aug-2004.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
Theorem	cnvresima 4837	An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)
Theorem	resdm2 4838	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
Theorem	resdmres 4839	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
Theorem	imadmres 4840	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)
Theorem	mptpreima 4841*	The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
Theorem	mptiniseg 4842*	Converse singleton image of a function defined by maps-to. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐶 ∈ 𝑉 → (◡𝐹 “ {𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐵 = 𝐶})
Theorem	dmmpt 4843	The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
Theorem	dmmptss 4844*	The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 ⊆ 𝐴
Theorem	dmmptg 4845*	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.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
Theorem	relco 4846	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
⊢ Rel (𝐴 ∘ 𝐵)
Theorem	dfco2 4847*	Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
Theorem	dfco2a 4848*	Generalization of dfco2 4847, 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 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
Theorem	coundi 4849	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))
Theorem	coundir 4850	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))
Theorem	cores 4851	Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (ran 𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ∘ 𝐵) = (𝐴 ∘ 𝐵))
Theorem	resco 4852	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))
Theorem	imaco 4853	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))
Theorem	rnco 4854	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rnco2 4855	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)
Theorem	dmco 4856	The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)
Theorem	coiun 4857*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	cocnvcnv1 4858	A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
⊢ (◡◡𝐴 ∘ 𝐵) = (𝐴 ∘ 𝐵)
Theorem	cocnvcnv2 4859	A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
⊢ (𝐴 ∘ ◡◡𝐵) = (𝐴 ∘ 𝐵)
Theorem	cores2 4860	Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
⊢ (dom 𝐴 ⊆ 𝐶 → (𝐴 ∘ ◡(◡𝐵 ↾ 𝐶)) = (𝐴 ∘ 𝐵))
Theorem	co02 4861	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
⊢ (𝐴 ∘ ∅) = ∅
Theorem	co01 4862	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
⊢ (∅ ∘ 𝐴) = ∅
Theorem	coi1 4863	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)
Theorem	coi2 4864	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)
Theorem	coires1 4865	Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)
Theorem	coass 4866	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.)
⊢ ((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))
Theorem	relcnvtr 4867	A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
Theorem	relssdmrn 4868	A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Theorem	cnvssrndm 4869	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)
Theorem	cossxp 4870	Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Theorem	relrelss 4871	Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
Theorem	unielrel 4872	The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∪ 𝐴 ∈ ∪ 𝑅)
Theorem	relfld 4873	The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Theorem	relresfld 4874	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
Theorem	relcoi2 4875	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅)
Theorem	relcoi1 4876	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)
Theorem	unidmrn 4877	The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴)
Theorem	relcnvfld 4878	if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
Theorem	dfdm2 4879	Alternate definition of domain df-dm 4382 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)
Theorem	unixpm 4880*	The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
Theorem	unixp0im 4881	The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
Theorem	cnvexg 4882	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ V)
Theorem	cnvex 4883	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ◡𝐴 ∈ V
Theorem	relcnvexb 4884	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
⊢ (Rel 𝑅 → (𝑅 ∈ V ↔ ◡𝑅 ∈ V))
Theorem	ressn 4885	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Theorem	cnviinm 4886*	The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
⊢ (∃𝑦 𝑦 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
Theorem	cnvpom 4887*	The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴))
Theorem	cnvsom 4888*	The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴))
Theorem	coexg 4889	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∘ 𝐵) ∈ V)
Theorem	coex 4890	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∘ 𝐵) ∈ V
Theorem	xpcom 4891*	Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
2.6.7  Definite description binder (inverted iota)
Syntax	cio 4892	Extend class notation with Russell's definition description binder (inverted iota).
class (℩𝑥𝜑)
Theorem	iotajust 4893*	Soundness justification theorem for df-iota 4894. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
Definition	df-iota 4894*	Define Russell's definition description binder, which can be read as "the unique 𝑥 such that 𝜑," where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see iotaval 4905); otherwise, it evaluates to the empty set (see iotanul 4909). Russell used the inverted iota symbol ℩ to represent the binder. Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 4917 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
Theorem	dfiota2 4895*	Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
Theorem	nfiota1 4896	Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥(℩𝑥𝜑)
Theorem	nfiotadxy 4897*	Deduction version of nfiotaxy 4898. (Contributed by Jim Kingdon, 21-Dec-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
Theorem	nfiotaxy 4898*	Bound-variable hypothesis builder for the ℩ class. (Contributed by NM, 23-Aug-2011.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥(℩𝑦𝜑)
Theorem	cbviota 4899	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)
Theorem	cbviotav 4900*	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)