P. 63 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dmresv 6201	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ dom (𝐴 ↾ V) = dom 𝐴
Theorem	rnresv 6202	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ ran (𝐴 ↾ V) = ran 𝐴
Theorem	dfrn4 6203	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
⊢ ran 𝐴 = (𝐴 “ V)
Theorem	imadifssran 6204	Condition for the range of a class to be the range of one of its restrictions. (Contributed by AV, 4-Oct-2025.) Remove antecedent. (Revised by Eric Schmidt, 10-Jul-2026.)
⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
Theorem	imadifssranOLD 6205	Obsolete version of imadifssran 6204 as of 10-Jul-2026. (Contributed by AV, 4-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
Theorem	csbrn 6206	Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵
Theorem	rescnvcnv 6207	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 6208	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
Theorem	imacnvcnv 6209	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)
Theorem	dmsnn0 6210	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} ≠ ∅)
Theorem	rnsnn0 6211	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} ≠ ∅)
Theorem	dmsn0 6212	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
⊢ dom {∅} = ∅
Theorem	cnvsn0 6213	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ◡{∅} = ∅
Theorem	dmsn0el 6214	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	relsn2 6215	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.)
⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))
Theorem	dmsnopg 6216	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	dmsnopss 6217	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}
Theorem	dmpropg 6218	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 6219	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 6220	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 6221	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 6222	Double converse of a singleton of an ordered pair. (Unlike cnvsn 6229, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}
Theorem	dmsnsnsn 6223	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}
Theorem	rnsnopg 6224	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 6225	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	cnvsng 6226	Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.) (Proof shortened by BJ, 12-Feb-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Theorem	rnsnop 6227	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 6228	Extract the first member of an ordered pair. (See op2nda 6231 to extract the second member, op1stb 5455 for an alternate version, and op1st 7995 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
Theorem	cnvsn 6229	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⟩}
Theorem	op2ndb 6230	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5455 to extract the first member, op2nda 6231 for an alternate version, and op2nd 7996 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵
Theorem	op2nda 6231	Extract the second member of an ordered pair. (See op1sta 6228 to extract the first member, op2ndb 6230 for an alternate version, and op2nd 7996 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
Theorem	opswap 6232	Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)
⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩
Theorem	cnvresima 6233	An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)
Theorem	resdm2 6234	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
Theorem	resdmres 6235	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
Theorem	resresdm 6236	A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
⊢ (𝐹 = (𝐸 ↾ 𝐴) → 𝐹 = (𝐸 ↾ dom 𝐹))
Theorem	imadmres 6237	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)
Theorem	resdmss 6238	Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
Theorem	resdifdi 6239	Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))
Theorem	resdifdir 6240	Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))
Theorem	mptpreima 6241*	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 6242*	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 6243	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 6244*	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 6245*	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	rnmpt0f 6246*	The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → (ran 𝐹 = ∅ ↔ 𝐴 = ∅))
Theorem	rnmptn0 6247*	The range of a function in maps-to notation is nonempty if the domain is nonempty. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ran 𝐹 ≠ ∅)
Theorem	dfco2 6248*	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 6249*	Generalization of dfco2 6248, 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 6250	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 6251	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 6252	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 6253	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))
Theorem	imaco 6254	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.)
⊢ ((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))
Theorem	rnco 6255	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) Avoid ax-11 2192. (Revised by TM, 24-Jan-2026.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rncoOLD 6256	Obsolete version of rnco 6255 as of 24-Jan-2026. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rnco2 6257	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)
Theorem	dmco 6258	The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
⊢ dom (𝐴 ∘ 𝐵) = (◡𝐵 “ dom 𝐴)
Theorem	coeq0 6259	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 6250 and coundir 6251 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
Theorem	coiun 6260*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	cocnvcnv1 6261	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 6262	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 6263	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 6264	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
⊢ (𝐴 ∘ ∅) = ∅
Theorem	co01 6265	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
⊢ (∅ ∘ 𝐴) = ∅
Theorem	coi1 6266	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 6267	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 6268	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 6269	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	relcnvtrg 6270	General form of relcnvtr 6271. (Contributed by Peter Mazsa, 17-Oct-2023.)
⊢ ((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇))
Theorem	relcnvtr 6271	A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
⊢ (Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
Theorem	relssdmrn 6272	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.) (Proof shortened by SN, 23-Dec-2024.)
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
Theorem	resssxp 6273	If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝑅 “ 𝐴) ⊆ 𝐵 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐵))
Theorem	cnvssrndm 6274	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 6275	Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
⊢ (𝐴 ∘ 𝐵) ⊆ (dom 𝐵 × ran 𝐴)
Theorem	relrelss 6276	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 6277	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 6278	The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
Theorem	relresfld 6279	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
Theorem	relcoi2 6280	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 6281	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)
Theorem	unidmrn 6282	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 6283	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 6284	Alternate definition of domain df-dm 5673 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴)
Theorem	unixp 6285	The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)
⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
Theorem	unixp0 6286	A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
Theorem	unixpid 6287	Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.)
⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴
Theorem	ressn 6288	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
Theorem	cnviin 6289*	The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
Theorem	cnvpo 6290	The converse of a partial order is a partial order. (Contributed by NM, 15-Jun-2005.)
⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)
Theorem	cnvso 6291	The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
Theorem	xpco 6292	Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Theorem	xpcoid 6293	Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
Theorem	elsnxp 6294*	Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Theorem	reu3op 6295*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 𝜒 ∧ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Theorem	reuop 6296*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 23-Jun-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜃))    ⇒   ⊢ (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑌 (𝜒 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝜃 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
Theorem	opreu2reurex 6297*	There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒))
Theorem	opreu2reu 6298*	If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎 ∈ 𝐴 ∃!𝑏 ∈ 𝐵 𝜒)
Theorem	dfpo2 6299	Quantifier-free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅))
Theorem	csbcog 6300	Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶))