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Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfima3 4701* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥(𝑥𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Theoremelimag 4702* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))

Theoremelima 4703* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)

Theoremelima2 4704* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))

Theoremelima3 4705* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

Theoremnfima 4706 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremnfimad 4707 Deduction version of bound-variable hypothesis builder nfima 4706. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐵))

Theoremimadmrn 4708 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
(𝐴 “ dom 𝐴) = ran 𝐴

Theoremimassrn 4709 The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.)
(𝐴𝐵) ⊆ ran 𝐴

Theoremimaexg 4710 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremimai 4711 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ 𝐴) = 𝐴

Theoremrnresi 4712 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ 𝐴) = 𝐴

Theoremresiima 4713 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)

Theoremima0 4714 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
(𝐴 “ ∅) = ∅

Theorem0ima 4715 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
(∅ “ 𝐴) = ∅

Theoremcsbima12g 4716 Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))

Theoremimadisj 4717 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)

Theoremcnvimass 4718 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
(𝐴𝐵) ⊆ dom 𝐴

Theoremcnvimarndm 4719 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 “ ran 𝐴) = dom 𝐴

Theoremimasng 4720* The image of a singleton. (Contributed by NM, 8-May-2005.)
(𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})

Theoremelreimasng 4721 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))

Theoremelimasn 4722 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐵 ∈ V    &   𝐶 ∈ V       (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

Theoremelimasng 4723 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Theoremargs 4724* Two ways to express the class of unique-valued arguments of 𝐹, which is the same as the domain of 𝐹 whenever 𝐹 is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg 𝐹 " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)
{𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Theoremeliniseg 4725 Membership in an initial segment. The idiom (𝐴 “ {𝐵}), meaning {𝑥𝑥𝐴𝐵}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐶 ∈ V       (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Theoremepini 4726 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝐴 ∈ V       ( E “ {𝐴}) = 𝐴

Theoreminiseg 4727* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
(𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})

Theoremdfse2 4728* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)

Theoremexse2 4729 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅𝑉𝑅 Se 𝐴)

Theoremimass1 4730 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremimass2 4731 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremndmima 4732 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)

Theoremrelcnv 4733 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
Rel 𝐴

Theoremrelbrcnvg 4734 When 𝑅 is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
(Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremrelbrcnv 4735 When 𝑅 is a relation, the sethood assumptions on brcnv 4546 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵𝑅𝐴)

Theoremcotr 4736* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremissref 4737* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
(( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)

Theoremcnvsym 4738* 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.)
(𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))

Theoremintasym 4739* 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.)
((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))

Theoremasymref 4740* Two ways of saying a relation is antisymmetric and reflexive. 𝑅 is the field of a relation by relfld 4876. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝑅𝑅) = ( I ↾ 𝑅) ↔ ∀𝑥 𝑅𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ 𝑥 = 𝑦))

Theoremintirr 4741* 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)

Theorembrcodir 4742* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧𝐵𝑅𝑧)))

Theoremcodir 4743* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))

Theoremqfto 4744* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑅𝑦𝑦𝑅𝑥))

Theoremxpidtr 4745 A square cross product (𝐴 × 𝐴) is a transitive relation. (Contributed by FL, 31-Jul-2009.)
((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)

Theoremtrin2 4746 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
(((𝑅𝑅) ⊆ 𝑅 ∧ (𝑆𝑆) ⊆ 𝑆) → ((𝑅𝑆) ∘ (𝑅𝑆)) ⊆ (𝑅𝑆))

Theorempoirr2 4747 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
(𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)

Theoremtrinxp 4748 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
((𝑅𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))

Theoremsoirri 4749 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

Theoremsotri 4750 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)       ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Theoremson2lpi 4751 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)        ¬ (𝐴𝑅𝐵𝐵𝑅𝐴)

Theoremsotri2 4752 A transitivity relation. (Read ¬ B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)       ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Theoremsotri3 4753 A transitivity relation. (Read A < B and ¬ C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
𝑅 Or 𝑆    &   𝑅 ⊆ (𝑆 × 𝑆)       ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Theorempoleloe 4754 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
(𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))

Theorempoltletr 4755 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝑅 Po 𝑋 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑅𝐵𝐵(𝑅 ∪ I )𝐶) → 𝐴𝑅𝐶))

Theoremcnvopab 4756* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}

Theoremcnv0 4757 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
∅ = ∅

Theoremcnvi 4758 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

Theoremcnvun 4759 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.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcnvdif 4760 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
(𝐴𝐵) = (𝐴𝐵)

Theoremcnvin 4761 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.)
(𝐴𝐵) = (𝐴𝐵)

Theoremrnun 4762 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
ran (𝐴𝐵) = (ran 𝐴 ∪ ran 𝐵)

Theoremrnin 4763 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
ran (𝐴𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

Theoremrniun 4764 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵

Theoremrnuni 4765* 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 = 𝑥𝐴 ran 𝑥

Theoremimaundi 4766 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremimaundir 4767 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremdminss 4768 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 𝑅𝐴) ⊆ (𝑅 “ (𝑅𝐴))

Theoremimainss 4769 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
((𝑅𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (𝑅𝐵)))

Theoreminimass 4770 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝐴𝐵) “ 𝐶) ⊆ ((𝐴𝐶) ∩ (𝐵𝐶))

Theoreminimasn 4771 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
(𝐶𝑉 → ((𝐴𝐵) “ {𝐶}) = ((𝐴 “ {𝐶}) ∩ (𝐵 “ {𝐶})))

Theoremcnvxp 4772 The converse of a cross 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)

Theoremxp0 4773 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
(𝐴 × ∅) = ∅

Theoremxpmlem 4774* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

Theoremxpm 4775* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

Theoremxpeq0r 4776 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)

Theoremxpdisj1 4777 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
((𝐴𝐵) = ∅ → ((𝐴 × 𝐶) ∩ (𝐵 × 𝐷)) = ∅)

Theoremxpdisj2 4778 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
((𝐴𝐵) = ∅ → ((𝐶 × 𝐴) ∩ (𝐷 × 𝐵)) = ∅)

Theoremxpsndisj 4779 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
(𝐵𝐷 → ((𝐴 × {𝐵}) ∩ (𝐶 × {𝐷})) = ∅)

Theoremdjudisj 4780* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((𝐴𝐵) = ∅ → ( 𝑥𝐴 ({𝑥} × 𝐶) ∩ 𝑦𝐵 ({𝑦} × 𝐷)) = ∅)

Theoremresdisj 4781 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴𝐵) = ∅ → ((𝐶𝐴) ↾ 𝐵) = ∅)

Theoremrnxpm 4782* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
(∃𝑥 𝑥𝐴 → ran (𝐴 × 𝐵) = 𝐵)

Theoremdmxpss 4783 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
dom (𝐴 × 𝐵) ⊆ 𝐴

Theoremrnxpss 4784 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
ran (𝐴 × 𝐵) ⊆ 𝐵

Theoremrnxpid 4785 The range of a square cross product. (Contributed by FL, 17-May-2010.)
ran (𝐴 × 𝐴) = 𝐴

Theoremssxpbm 4786* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
(∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷)))

Theoremssxp1 4787* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
(∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) ⊆ (𝐵 × 𝐶) ↔ 𝐴𝐵))

Theoremssxp2 4788* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
(∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) ⊆ (𝐶 × 𝐵) ↔ 𝐴𝐵))

Theoremxp11m 4789* The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)
((∃𝑥 𝑥𝐴 ∧ ∃𝑦 𝑦𝐵) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremxpcanm 4790* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
(∃𝑥 𝑥𝐶 → ((𝐶 × 𝐴) = (𝐶 × 𝐵) ↔ 𝐴 = 𝐵))

Theoremxpcan2m 4791* Cancellation law for cross-product. (Contributed by Jim Kingdon, 14-Dec-2018.)
(∃𝑥 𝑥𝐶 → ((𝐴 × 𝐶) = (𝐵 × 𝐶) ↔ 𝐴 = 𝐵))

Theoremxpexr2m 4792* If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)
(((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theoremssrnres 4793 Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
(𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Theoremrninxp 4794* Range of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑥𝐶𝑦)

Theoremdminxp 4795* Domain of the intersection with a cross product. (Contributed by NM, 17-Jan-2006.)
(dom (𝐶 ∩ (𝐴 × 𝐵)) = 𝐴 ↔ ∀𝑥𝐴𝑦𝐵 𝑥𝐶𝑦)

Theoremimainrect 4796 Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.)
((𝐺 ∩ (𝐴 × 𝐵)) “ 𝑌) = ((𝐺 “ (𝑌𝐴)) ∩ 𝐵)

Theoremxpima1 4797 The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝐴𝐶) = ∅ → ((𝐴 × 𝐵) “ 𝐶) = ∅)

Theoremxpima2m 4798* The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
(∃𝑥 𝑥 ∈ (𝐴𝐶) → ((𝐴 × 𝐵) “ 𝐶) = 𝐵)

Theoremxpimasn 4799 The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
(𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Theoremcnvcnv3 4800* 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}

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