P. 62 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnvxp 6101	The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
Theorem	xp0 6102	The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
⊢ (𝐴 × ∅) = ∅
Theorem	xpnz 6103	The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
Theorem	xpeq0 6104	At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
Theorem	xpdisj1 6105	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Theorem	xpdisj2 6106	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Theorem	xpsndisj 6107	Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
Theorem	difxp 6108	Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.) (Proof shortened by Wolf Lammen, 16-May-2025.)
⊢ ((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))
Theorem	difxp1 6109	Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐴 ∖ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶))
Theorem	difxp2 6110	Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))
Theorem	djudisj 6111*	Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝐷)) = ∅)
Theorem	xpdifid 6112*	The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )
Theorem	resdisj 6113	A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 ↾ 𝐴) ↾ 𝐵) = ∅)
Theorem	rnxp 6114	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)
Theorem	dmxpss 6115	The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)
⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
Theorem	rnxpss 6116	The range of a Cartesian product is included in its second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
Theorem	rnxpid 6117	The range of a Cartesian square. (Contributed by FL, 17-May-2010.)
⊢ ran (𝐴 × 𝐴) = 𝐴
Theorem	ssxpb 6118	A Cartesian product subclass relationship is equivalent to the conjunction of the analogous relationships for the factors. (Contributed by NM, 17-Dec-2008.)
⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
Theorem	xp11 6119	The Cartesian product of nonempty classes is a one-to-one "function" of its two "arguments". In other words, two Cartesian products, at least one with nonempty factors, are equal if and only if their respective factors are equal. (Contributed by NM, 31-May-2008.)
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	xpcan 6120	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xpcan2 6121	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Theorem	ssrnres 6122	Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product): the LHS expresses inclusion in the range of the restricted relation, while the RHS expresses equality with the range of the restricted and corestricted relation. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Theorem	rninxp 6123*	Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
Theorem	dminxp 6124*	Two ways to express totality of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.)
⊢ (dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥𝐶𝑦)
Theorem	imainrect 6125	Image by a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by Stefan O'Rear, 19-Feb-2015.)
⊢ ((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌 ∩ 𝐴)) ∩ 𝐵)
Theorem	xpima 6126	Direct image by a Cartesian product. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
Theorem	xpima1 6127	Direct image by a Cartesian product (case of empty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ ((𝐴 ∩ 𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)
Theorem	xpima2 6128	Direct image by a Cartesian product (case of nonempty intersection with the domain). (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ ((𝐴 ∩ 𝐶) ≠ ∅ → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)
Theorem	xpimasn 6129	Direct image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.)
⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Theorem	sossfld 6130	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 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ⊆ (dom 𝑅 ∪ ran 𝑅))
Theorem	sofld 6131	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 𝐴 ∧ 𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))
Theorem	cnvcnv3 6132*	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}
Theorem	dfrel2 6133	Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
Theorem	dfrel4v 6134*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6875 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Theorem	dfrel4 6135*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6875 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑦𝑅    ⇒   ⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Theorem	cnvcnv 6136	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))
Theorem	cnvcnv2 6137	The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
⊢ ◡◡𝐴 = (𝐴 ↾ V)
Theorem	cnvcnvss 6138	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
⊢ ◡◡𝐴 ⊆ 𝐴
Theorem	cnvrescnv 6139	Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))
Theorem	cnveqb 6140	Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))
Theorem	cnveq0 6141	A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))
Theorem	dfrel3 6142	Alternate definition of relation. (Contributed by NM, 14-May-2008.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
Theorem	elid 6143*	Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6142 after elrid 5992. (Contributed by BJ, 28-Aug-2022.)
⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
Theorem	dmresv 6144	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ dom (𝐴 ↾ V) = dom 𝐴
Theorem	rnresv 6145	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ ran (𝐴 ↾ V) = ran 𝐴
Theorem	dfrn4 6146	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
⊢ ran 𝐴 = (𝐴 “ V)
Theorem	csbrn 6147	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 6148	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 6149	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
Theorem	imacnvcnv 6150	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)
Theorem	dmsnn0 6151	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 6152	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 6153	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
⊢ dom {∅} = ∅
Theorem	cnvsn0 6154	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ◡{∅} = ∅
Theorem	dmsn0el 6155	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 6156	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 6157	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 6158	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 6159	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 6160	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 6161	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 6162	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 6163	Double converse of a singleton of an ordered pair. (Unlike cnvsn 6170, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}
Theorem	dmsnsnsn 6164	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 6165	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 6166	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 6167	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 6168	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 6169	Extract the first member of an ordered pair. (See op2nda 6172 to extract the second member, op1stb 5409 for an alternate version, and op1st 7924 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
Theorem	cnvsn 6170	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 6171	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5409 to extract the first member, op2nda 6172 for an alternate version, and op2nd 7925 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵
Theorem	op2nda 6172	Extract the second member of an ordered pair. (See op1sta 6169 to extract the first member, op2ndb 6171 for an alternate version, and op2nd 7925 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 6173	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 6174	An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)
⊢ (◡(𝐹 ↾ 𝐴) “ 𝐵) = ((◡𝐹 “ 𝐵) ∩ 𝐴)
Theorem	resdm2 6175	A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
Theorem	resdmres 6176	Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
Theorem	resresdm 6177	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 6178	The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵)
Theorem	resdmss 6179	Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.)
⊢ dom (𝐴 ↾ 𝐵) ⊆ 𝐵
Theorem	resdifdi 6180	Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))
Theorem	resdifdir 6181	Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))
Theorem	mptpreima 6182*	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 6183*	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 6184	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 6185*	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 6186*	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 6187*	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 6188*	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 6189*	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 6190*	Generalization of dfco2 6189, 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 6191	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 6192	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 6193	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 6194	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
⊢ ((𝐴 ∘ 𝐵) ↾ 𝐶) = (𝐴 ∘ (𝐵 ↾ 𝐶))
Theorem	imaco 6195	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 6196	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
Theorem	rnco2 6197	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
⊢ ran (𝐴 ∘ 𝐵) = (𝐴 “ ran 𝐵)
Theorem	dmco 6198	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 6199	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 6191 and coundir 6192 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
Theorem	coiun 6200*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)