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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreseq12d 4901 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfres 4902 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremcsbresg 4903 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
Theoremres0 4904 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
(𝐴 ↾ ∅) = ∅
 
Theoremopelres 4905 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
 
Theorembrres 4906 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
𝐵 ∈ V       (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷))
 
Theoremopelresg 4907 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
(𝐵𝑉 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷)))
 
Theorembrresg 4908 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
(𝐵𝑉 → (𝐴(𝐶𝐷)𝐵 ↔ (𝐴𝐶𝐵𝐴𝐷)))
 
Theoremopres 4909 Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
𝐵 ∈ V       (𝐴𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
 
Theoremresieq 4910 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
((𝐵𝐴𝐶𝐴) → (𝐵( I ↾ 𝐴)𝐶𝐵 = 𝐶))
 
Theoremopelresi 4911 𝐴, 𝐴 belongs to a restriction of the identity class iff 𝐴 belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
(𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
 
Theoremresres 4912 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
 
Theoremresundi 4913 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremresundir 4914 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremresindi 4915 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
(𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoremresindir 4916 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoreminres 4917 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ↾ 𝐶)
 
Theoremresdifcom 4918 Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 
Theoremresiun1 4919* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
 
Theoremresiun2 4920* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
(𝐶 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶𝐵)
 
Theoremdmres 4921 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
 
Theoremssdmres 4922 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
(𝐴 ⊆ dom 𝐵 ↔ dom (𝐵𝐴) = 𝐴)
 
Theoremdmresexg 4923 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
(𝐵𝑉 → dom (𝐴𝐵) ∈ V)
 
Theoremresss 4924 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵) ⊆ 𝐴
 
Theoremrescom 4925 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 
Theoremssres 4926 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssres2 4927 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremrelres 4928 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Rel (𝐴𝐵)
 
Theoremresabs1 4929 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
(𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 
Theoremresabs1d 4930 Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐶)       (𝜑 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 
Theoremresabs2 4931 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
(𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 
Theoremresidm 4932 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
 
Theoremresima 4933 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
((𝐴𝐵) “ 𝐵) = (𝐴𝐵)
 
Theoremresima2 4934 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
(𝐵𝐶 → ((𝐴𝐶) “ 𝐵) = (𝐴𝐵))
 
Theoremxpssres 4935 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐶𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵))
 
Theoremelres 4936* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
(𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
 
Theoremelsnres 4937* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
𝐶 ∈ V       (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
 
Theoremrelssres 4938 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
((Rel 𝐴 ∧ dom 𝐴𝐵) → (𝐴𝐵) = 𝐴)
 
Theoremresdm 4939 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
(Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
 
Theoremresexg 4940 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoremresex 4941 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremresindm 4942 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
(Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴𝐵))
 
Theoremresdmdfsn 4943 Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.)
(Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))
 
Theoremresopab 4944* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
 
Theoremresiexg 4945 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
(𝐴𝑉 → ( I ↾ 𝐴) ∈ V)
 
Theoremiss 4946 A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ⊆ I ↔ 𝐴 = ( I ↾ dom 𝐴))
 
Theoremresopab2 4947* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
(𝐴𝐵 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝜑)} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)})
 
Theoremresmpt 4948* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
(𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremresmpt3 4949* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
 
Theoremresmptf 4950 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremresmptd 4951* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐵𝐴)       (𝜑 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 
Theoremdfres2 4952* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
(𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑥𝑅𝑦)}
 
Theoremopabresid 4953* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝑥)} = ( I ↾ 𝐴)
 
Theoremmptresid 4954* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)
(𝑥𝐴𝑥) = ( I ↾ 𝐴)
 
Theoremdmresi 4955 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
dom ( I ↾ 𝐴) = 𝐴
 
Theoremrestidsing 4956 Restriction of the identity to a singleton. (Contributed by FL, 2-Aug-2009.) (Proof shortened by JJ, 25-Aug-2021.) (Proof shortened by Peter Mazsa, 6-Oct-2022.)
( I ↾ {𝐴}) = ({𝐴} × {𝐴})
 
Theoremresid 4957 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
(Rel 𝐴 → (𝐴 ↾ V) = 𝐴)
 
Theoremimaeq1 4958 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2 4959 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq1i 4960 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremimaeq2i 4961 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremimaeq1d 4962 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremimaeq2d 4963 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremimaeq12d 4964 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdfima2 4965* Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝐵) = {𝑦 ∣ ∃𝑥𝐵 𝑥𝐴𝑦}
 
Theoremdfima3 4966* 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 4967* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
(𝐴𝑉 → (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴))
 
Theoremelima 4968* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥𝐶 𝑥𝐵𝐴)
 
Theoremelima2 4969* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
𝐴 ∈ V       (𝐴 ∈ (𝐵𝐶) ↔ ∃𝑥(𝑥𝐶𝑥𝐵𝐴))
 
Theoremelima3 4970* 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 4971 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 4972 Deduction version of bound-variable hypothesis builder nfima 4971. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥(𝐴𝐵))
 
Theoremimadmrn 4973 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 4974 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 4975 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)
 
Theoremimaex 4976 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V
 
Theoremimai 4977 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
( I “ 𝐴) = 𝐴
 
Theoremrnresi 4978 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
ran ( I ↾ 𝐴) = 𝐴
 
Theoremresiima 4979 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
(𝐵𝐴 → (( I ↾ 𝐴) “ 𝐵) = 𝐵)
 
Theoremima0 4980 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
(𝐴 “ ∅) = ∅
 
Theorem0ima 4981 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
(∅ “ 𝐴) = ∅
 
Theoremcsbima12g 4982 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 4983 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 4984 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
(𝐴𝐵) ⊆ dom 𝐴
 
Theoremcnvimarndm 4985 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 4986* The image of a singleton. (Contributed by NM, 8-May-2005.)
(𝐴𝐵 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
 
Theoremelreimasng 4987 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
((Rel 𝑅𝐴𝑉) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 𝐴𝑅𝐵))
 
Theoremelimasn 4988 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 4989 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
 
Theoremargs 4990* 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 4991 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 4992 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 4993* 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 4994* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
 
Theoremexse2 4995 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
(𝑅𝑉𝑅 Se 𝐴)
 
Theoremimass1 4996 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremimass2 4997 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremndmima 4998 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
𝐴 ∈ dom 𝐵 → (𝐵 “ {𝐴}) = ∅)
 
Theoremrelcnv 4999 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
Rel 𝐴
 
Theoremrelbrcnvg 5000 When 𝑅 is a relation, the sethood assumptions on brcnv 4803 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
(Rel 𝑅 → (𝐴𝑅𝐵𝐵𝑅𝐴))
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