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	son2lpi 6101	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 6102	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 6103	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 6104	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 6105	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 6106	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 6107	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 6108	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 6109	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 6110*	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 2142, ax-12 2178. (Revised by SN, 7-Jun-2025.)
⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Theorem	cnvopabOLD 6111*	Obsolete version of cnvopab 6110 as of 7-Jun-2025. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
Theorem	mptcnv 6112*	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	cnv0 6113	The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5251, ax-nul 5261, ax-pr 5387. (Revised by KP, 25-Oct-2021.)
⊢ ◡∅ = ∅
Theorem	cnvi 6114	The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡ I = I
Theorem	cnvun 6115	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 6116	Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ◡(𝐴 ∖ 𝐵) = (◡𝐴 ∖ ◡𝐵)
Theorem	cnvin 6117	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 6118	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 6119	The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)
Theorem	rniun 6120	The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵
Theorem	rnuni 6121*	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 6122	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 6123	The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))
Theorem	imadifssran 6124	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 6125	The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6054 for subsets. (Contributed by AV, 18-Sep-2024.)
⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
Theorem	dminss 6126	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 6127	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 6128	The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)
⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))
Theorem	inimasn 6129	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 6130	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 6131	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 6132	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 6133	At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.)
⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
Theorem	xpdisj1 6134	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)
Theorem	xpdisj2 6135	Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)
Theorem	xpsndisj 6136	Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
⊢ (𝐵 ≠ 𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)
Theorem	difxp 6137	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 6138	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 6139	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 6140*	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 6141*	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 6142	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 6143	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 6144	The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)
⊢ dom (𝐴 × 𝐵) ⊆ 𝐴
Theorem	rnxpss 6145	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 6146	The range of a Cartesian square. (Contributed by FL, 17-May-2010.)
⊢ ran (𝐴 × 𝐴) = 𝐴
Theorem	ssxpb 6147	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 6148	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 6149	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))
Theorem	xpcan2 6150	Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.)
⊢ (𝐶 ≠ ∅ → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))
Theorem	ssrnres 6151	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 6152*	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 6153*	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 6154	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 6155	Direct image by a Cartesian product. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ ((𝐴 × 𝐵) “ 𝐶) = if((𝐴 ∩ 𝐶) = ∅, ∅, 𝐵)
Theorem	xpima1 6156	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 6157	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 6158	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 6159	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 6160	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 6161*	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 6162	Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
Theorem	dfrel4v 6163*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6919 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.)
⊢ (Rel 𝑅 ↔ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦})
Theorem	dfrel4 6164*	A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6919 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 6165	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 6166	The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
⊢ ◡◡𝐴 = (𝐴 ↾ V)
Theorem	cnvcnvss 6167	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 6168	Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
⊢ ◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))
Theorem	cnveqb 6169	Equality theorem for converse. (Contributed by FL, 19-Sep-2011.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ◡𝐴 = ◡𝐵))
Theorem	cnveq0 6170	A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))
Theorem	dfrel3 6171	Alternate definition of relation. (Contributed by NM, 14-May-2008.)
⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
Theorem	elid 6172*	Characterization of the elements of the identity relation. TODO: reorder theorems to move this theorem and dfrel3 6171 after elrid 6017. (Contributed by BJ, 28-Aug-2022.)
⊢ (𝐴 ∈ I ↔ ∃𝑥 𝐴 = ⟨𝑥, 𝑥⟩)
Theorem	dmresv 6173	The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ dom (𝐴 ↾ V) = dom 𝐴
Theorem	rnresv 6174	The range of a universal restriction. (Contributed by NM, 14-May-2008.)
⊢ ran (𝐴 ↾ V) = ran 𝐴
Theorem	dfrn4 6175	Range defined in terms of image. (Contributed by NM, 14-May-2008.)
⊢ ran 𝐴 = (𝐴 “ V)
Theorem	csbrn 6176	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 6177	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 6178	The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.)
⊢ ◡◡(𝐴 ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
Theorem	imacnvcnv 6179	The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵)
Theorem	dmsnn0 6180	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 6181	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 6182	The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)
⊢ dom {∅} = ∅
Theorem	cnvsn0 6183	The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ◡{∅} = ∅
Theorem	dmsn0el 6184	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 6185	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 6186	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 6187	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 6188	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 6189	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 6190	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 6191	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 6192	Double converse of a singleton of an ordered pair. (Unlike cnvsn 6199, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}
Theorem	dmsnsnsn 6193	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 6194	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 6195	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 6196	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 6197	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 6198	Extract the first member of an ordered pair. (See op2nda 6201 to extract the second member, op1stb 5431 for an alternate version, and op1st 7976 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
Theorem	cnvsn 6199	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 6200	Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5431 to extract the first member, op2nda 6201 for an alternate version, and op2nd 7977 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵