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	relco 6101	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
⊢ Rel (𝐴 ∘ 𝐵)
Theorem	cotrg 6102*	Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6103 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6103. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2194. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
Theorem	cotr 6103*	Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 6102. (Contributed by NM, 27-Dec-1996.)
⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Theorem	idrefALT 6104*	Alternate proof of idref 7132 not relying on definitions related to functions. Two ways to state that a relation is reflexive on a class. (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Revised by NM, 30-Mar-2016.) (Proof shortened by BJ, 28-Aug-2022.) The "proof modification is discouraged" tag is here only because this is an *ALT result. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)
Theorem	cnvsym 6105*	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2194. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
Theorem	intasym 6106*	Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
Theorem	asymref 6107*	Two ways of saying a relation is antisymmetric and reflexive. ∪ ∪ 𝑅 is the field of a relation by relfld 6266. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ ∀𝑥 ∈ ∪ ∪ 𝑅∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))
Theorem	asymref2 6108*	Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
⊢ ((𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅) ↔ (∀𝑥 ∈ ∪ ∪ 𝑅𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦)))
Theorem	intirr 6109*	Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
Theorem	brcodir 6110*	Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
Theorem	codir 6111*	Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))
Theorem	qfto 6112*	A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
Theorem	xpidtr 6113	A Cartesian square is a transitive relation. (Contributed by FL, 31-Jul-2009.)
⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
Theorem	trin2 6114	The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
⊢ (((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))
Theorem	poirr2 6115	A partial order is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
Theorem	trinxp 6116	The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a Cartesian square is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
Theorem	soirri 6117	A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ 𝑅 Or 𝑆    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    ⇒   ⊢ ¬ 𝐴𝑅𝐴
Theorem	sotri 6118	A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ 𝑅 Or 𝑆    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    ⇒   ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	son2lpi 6119	A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
⊢ 𝑅 Or 𝑆    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    ⇒   ⊢ ¬ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴)
Theorem	sotri2 6120	A transitivity relation. (Read 𝐴 ≤ 𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
⊢ 𝑅 Or 𝑆    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    ⇒   ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	sotri3 6121	A transitivity relation. (Read 𝐴 < 𝐵 and 𝐵 ≤ 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
⊢ 𝑅 Or 𝑆    &   ⊢ 𝑅 ⊆ (𝑆 × 𝑆)    ⇒   ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
Theorem	poleloe 6122	Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)))
Theorem	poltletr 6123	Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝑅 Po 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑅𝐵 ∧ 𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))
Theorem	somin1 6124	Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐴)
Theorem	somincom 6125	Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
Theorem	somin2 6126	Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)
Theorem	soltmin 6127	Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐵, 𝐶) ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑅𝐶)))
Theorem	cnvopab 6128*	The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2178, ax-12 2215. (Revised by SN, 7-Jun-2025.)
⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Theorem	mptcnv 6129*	The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐷)))    ⇒   ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐶 ↦ 𝐷))
Theorem	cnvun 6130	The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡(𝐴 ∪ 𝐵) = (◡𝐴 ∪ ◡𝐵)
Theorem	cnvdif 6131	Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
Theorem	cnvin 6132	Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)
Theorem	rnun 6133	Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
⊢ ran (𝐴 ∪ 𝐵) = (ran 𝐴 ∪ ran 𝐵)
Theorem	rnin 6134	The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) Avoid ax-pr 5395 and ax-sep 5251. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Theorem	rninOLD 6135	Obsolete version of rnin 6134 as of 10-Jun-2026. (Contributed by NM, 15-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Theorem	rniun 6136	The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵
Theorem	rnuni 6137*	The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)
⊢ ran ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ran 𝑥
Theorem	imaundi 6138	Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))
Theorem	imaundir 6139	The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))
Theorem	imadifssran 6140	Condition for the range of a relation to be the range of one its restrictions. (Contributed by AV, 4-Oct-2025.)
⊢ ((Rel 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
Theorem	cnvimassrndm 6141	The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6076 for subsets. (Contributed by AV, 18-Sep-2024.)
⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
Theorem	dminss 6142	An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004.)
⊢ (dom 𝑅 ∩ 𝐴) ⊆ (◡𝑅 “ (𝑅 “ 𝐴))
Theorem	imainss 6143	An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))
Theorem	inimass 6144	The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))
Theorem	inimasn 6145	The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ (𝐶 ∈ 𝑉 → ((𝐴 ∩ 𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))
Theorem	cnvxp 6146	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	xp0OLD 6147	Obsolete version of xp0 5752 as of 1-Feb-2026. (Contributed by NM, 12-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 × ∅) = ∅
Theorem	xpnz 6148	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 6149	At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
Theorem	xpdisj1 6150	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Theorem	xpdisj2 6151	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Theorem	xpsndisj 6152	Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
Theorem	difxp 6153	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 6154	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 6155	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 6156*	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 6157*	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	xpdifcnvepel 6158*	The set of couples in a Cartesian product, where the second is not an element of the first. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐵 ∖ 𝑥)) = ((𝐴 × 𝐵) ∖ ◡ E )
Theorem	resdisj 6159	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 6160	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 6161	The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)
⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
Theorem	rnxpss 6162	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 6163	The range of a Cartesian square. (Contributed by FL, 17-May-2010.)
⊢ ran (𝐴 × 𝐴) = 𝐴
Theorem	ssxpb 6164	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 6165	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 6166	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xpcan2 6167	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Theorem	ssrnres 6168	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 6169*	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 6170*	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 6171	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 6172	Direct image by a Cartesian product. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
Theorem	xpima1 6173	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 6174	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 6175	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 6176	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 6177	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 6178*	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 6179	Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
Theorem	dfrel4v 6180*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6929 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Theorem	dfrel4 6181*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6929 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 6182	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 6183	The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
⊢ ◡◡𝐴 = (𝐴 ↾ V)
Theorem	cnvcnvss 6184	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ◡◡𝐴 ⊆ 𝐴
Theorem	cnvcnvssOLD 6185	Obsolete version of cnvcnvss 6184 as of 10-Jun-2026. (Contributed by NM, 23-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ◡◡𝐴 ⊆ 𝐴
Theorem	cnvrescnv 6186	Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))
Theorem	cnveqb 6187	Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))
Theorem	cnveq0 6188	A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))
Theorem	dfrel3 6189	Alternate definition of relation. (Contributed by NM, 14-May-2008.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
Theorem	elid 6190*	Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6189 after elrid 6039. (Contributed by BJ, 28-Aug-2022.)
⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
Theorem	dmresv 6191	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ dom (𝐴 ↾ V) = dom 𝐴
Theorem	rnresv 6192	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ ran (𝐴 ↾ V) = ran 𝐴
Theorem	dfrn4 6193	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
⊢ ran 𝐴 = (𝐴 “ V)
Theorem	csbrn 6194	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 6195	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 6196	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
Theorem	imacnvcnv 6197	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)
Theorem	dmsnn0 6198	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 6199	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 6200	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
⊢ dom {∅} = ∅