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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpssres 4901 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( C  C_  A  ->  ( ( A  X.  B )  |`  C )  =  ( C  X.  B ) )
 
Theoremelres 4902* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
 |-  ( A  e.  ( B  |`  C )  <->  E. x  e.  C  E. y ( A  =  <. x ,  y >.  /\ 
 <. x ,  y >.  e.  B ) )
 
Theoremelsnres 4903* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
 |-  C  e.  _V   =>    |-  ( A  e.  ( B  |`  { C } )  <->  E. y ( A  =  <. C ,  y >.  /\  <. C ,  y >.  e.  B ) )
 
Theoremrelssres 4904 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 4905 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 4906 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  ( A  |`  B )  e.  _V )
 
Theoremresex 4907 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresindm 4908 When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)
 |-  ( Rel  A  ->  ( A  |`  ( B  i^i  dom  A ) )  =  ( A  |`  B ) )
 
Theoremresdmdfsn 4909 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 4910* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
 |-  ( { <. x ,  y >.  |  ph }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
 
Theoremresiexg 4911 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4912 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  C_  _I  <->  A  =  (  _I  |`  dom  A )
 )
 
Theoremresopab2 4913* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
 |-  ( A  C_  B  ->  ( { <. x ,  y >.  |  ( x  e.  B  /\  ph ) }  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) } )
 
Theoremresmpt 4914* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
 |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremresmpt3 4915* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
 |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  ( A  i^i  B )  |->  C )
 
Theoremresmptf 4916 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremresmptd 4917* Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  B  C_  A )   =>    |-  ( ph  ->  (
 ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremdfres2 4918* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 4919* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 4920* The restricted identity expressed with the maps-to notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 4921 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 4922 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 4923 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 4924 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 4925 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 4926 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 4927 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 4928 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 4929 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 4930* 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.)
 |-  ( A " B )  =  { y  |  E. x  e.  B  x A y }
 
Theoremdfima3 4931* 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.)
 |-  ( A " B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A ) }
 
Theoremelimag 4932* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
 |-  ( A  e.  V  ->  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A ) )
 
Theoremelima 4933* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x  e.  C  x B A )
 
Theoremelima2 4934* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x ( x  e.  C  /\  x B A ) )
 
Theoremelima3 4935* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ( B " C )  <->  E. x ( x  e.  C  /\  <. x ,  A >.  e.  B ) )
 
Theoremnfima 4936 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A
 " B )
 
Theoremnfimad 4937 Deduction version of bound-variable hypothesis builder nfima 4936. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x ( A " B ) )
 
Theoremimadmrn 4938 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 4939 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.)
 |-  ( A " B )  C_  ran  A
 
Theoremimaexg 4940 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
 |-  ( A  e.  V  ->  ( A " B )  e.  _V )
 
Theoremimaex 4941 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by JJ, 24-Sep-2021.)
 |-  A  e.  _V   =>    |-  ( A " B )  e.  _V
 
Theoremimai 4942 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 4943 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 4944 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 4945 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 4946 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremcsbima12g 4947 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.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F "
 [_ A  /  x ]_ B ) )
 
Theoremimadisj 4948 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( A " B )  =  (/)  <->  ( dom  A  i^i  B )  =  (/) )
 
Theoremcnvimass 4949 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
 |-  ( `' A " B )  C_  dom  A
 
Theoremcnvimarndm 4950 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 4951* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelreimasng 4952 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
 |-  ( ( Rel  R  /\  A  e.  V ) 
 ->  ( B  e.  ( R " { A }
 ) 
 <->  A R B ) )
 
Theoremelimasn 4953 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( C  e.  ( A " { B }
 ) 
 <-> 
 <. B ,  C >.  e.  A )
 
Theoremelimasng 4954 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
 |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
 
Theoremargs 4955* 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  |  E. y ( F " { x } )  =  { y } }  =  { x  |  E! y  x F y }
 
Theoremeliniseg 4956 Membership in an initial segment. The idiom  ( `' A " { B } ), meaning  { x  |  x A B }, 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  e.  _V   =>    |-  ( B  e.  V  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
 
Theoremepini 4957 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
 |-  A  e.  _V   =>    |-  ( `'  _E  " { A } )  =  A
 
Theoreminiseg 4958* 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.)
 |-  ( B  e.  V  ->  ( `' A " { B } )  =  { x  |  x A B } )
 
Theoremdfse2 4959* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
 |-  ( R Se  A  <->  A. x  e.  A  ( A  i^i  ( `' R " { x } ) )  e. 
 _V )
 
Theoremexse2 4960 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 4961 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 4962 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ( C " A )  C_  ( C " B ) )
 
Theoremndmima 4963 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
 
Theoremrelcnv 4964 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
 |- 
 Rel  `' A
 
Theoremrelbrcnvg 4965 When  R is a relation, the sethood assumptions on brcnv 4769 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( Rel  R  ->  ( A `' R B  <->  B R A ) )
 
Theoremrelbrcnv 4966 When  R is a relation, the sethood assumptions on brcnv 4769 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcotr 4967* 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.)
 |-  ( ( R  o.  R )  C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
 
Theoremissref 4968* 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 ) 
 C_  R  <->  A. x  e.  A  x R x )
 
Theoremcnvsym 4969* 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.)
 |-  ( `' R  C_  R 
 <-> 
 A. x A. y
 ( x R y 
 ->  y R x ) )
 
Theoremintasym 4970* 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  i^i  `' R )  C_  _I  <->  A. x A. y
 ( ( x R y  /\  y R x )  ->  x  =  y ) )
 
Theoremasymref 4971* Two ways of saying a relation is antisymmetric and reflexive.  U. U. R is the field of a relation by relfld 5114. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) 
 <-> 
 A. x  e.  U. U. R A. y ( ( x R y 
 /\  y R x )  <->  x  =  y
 ) )
 
Theoremintirr 4972* 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^i  _I  )  =  (/)  <->  A. x  -.  x R x )
 
Theorembrcodir 4973* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( `' R  o.  R ) B  <->  E. z ( A R z  /\  B R z ) ) )
 
Theoremcodir 4974* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( A  X.  B )  C_  ( `' R  o.  R )  <->  A. x  e.  A  A. y  e.  B  E. z ( x R z  /\  y R z ) )
 
Theoremqfto 4975* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( A  X.  B )  C_  ( R  u.  `' R )  <->  A. x  e.  A  A. y  e.  B  ( x R y  \/  y R x ) )
 
Theoremxpidtr 4976 A square cross product  ( A  X.  A
) is a transitive relation. (Contributed by FL, 31-Jul-2009.)
 |-  ( ( A  X.  A )  o.  ( A  X.  A ) ) 
 C_  ( A  X.  A )
 
Theoremtrin2 4977 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
 |-  ( ( ( R  o.  R )  C_  R  /\  ( S  o.  S )  C_  S ) 
 ->  ( ( R  i^i  S )  o.  ( R  i^i  S ) ) 
 C_  ( R  i^i  S ) )
 
Theorempoirr2 4978 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
 
Theoremtrinxp 4979 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.)
 |-  ( ( R  o.  R )  C_  R  ->  ( ( R  i^i  ( A  X.  A ) )  o.  ( R  i^i  ( A  X.  A ) ) )  C_  ( R  i^i  ( A  X.  A ) ) )
 
Theoremsoirri 4980 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |- 
 -.  A R A
 
Theoremsotri 4981 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  C_  ( S  X.  S )   =>    |-  ( ( A R B  /\  B R C )  ->  A R C )
 
Theoremson2lpi 4982 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  C_  ( S  X.  S )   =>    |- 
 -.  ( A R B  /\  B R A )
 
Theoremsotri2 4983 A transitivity relation. (Read 
-. B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  ( ( A  e.  S  /\  -.  B R A  /\  B R C )  ->  A R C )
 
Theoremsotri3 4984 A transitivity relation. (Read A < B and  -. C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
 |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  ( ( C  e.  S  /\  A R B  /\  -.  C R B )  ->  A R C )
 
Theorempoleloe 4985 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( B  e.  V  ->  ( A ( R  u.  _I  ) B  <-> 
 ( A R B  \/  A  =  B ) ) )
 
Theorempoltletr 4986 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( ( A R B  /\  B ( R  u.  _I  ) C )  ->  A R C ) )
 
Theoremcnvopab 4987* 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 >.  |  ph }  =  { <. y ,  x >.  |  ph }
 
Theoremmptcnv 4988* The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  y  =  B ) 
 <->  ( y  e.  C  /\  x  =  D ) ) )   =>    |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  (
 y  e.  C  |->  D ) )
 
Theoremcnv0 4989 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 |-  `' (/)  =  (/)
 
Theoremcnvi 4990 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 4991 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  u.  B )  =  ( `' A  u.  `' B )
 
Theoremcnvdif 4992 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 4993 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  i^i  B )  =  ( `' A  i^i  `' B )
 
Theoremrnun 4994 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)
 |- 
 ran  ( A  u.  B )  =  ( ran  A  u.  ran  B )
 
Theoremrnin 4995 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
 |- 
 ran  ( A  i^i  B )  C_  ( ran  A  i^i  ran  B )
 
Theoremrniun 4996 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |- 
 ran  U_ x  e.  A  B  =  U_ x  e.  A  ran  B
 
Theoremrnuni 4997* 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  U. A  =  U_ x  e.  A  ran  x
 
Theoremimaundi 4998 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |-  ( A " ( B  u.  C ) )  =  ( ( A
 " B )  u.  ( A " C ) )
 
Theoremimaundir 4999 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 |-  ( ( A  u.  B ) " C )  =  ( ( A " C )  u.  ( B " C ) )
 
Theoremdminss 5000 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  i^i  A )  C_  ( `' R " ( R " A ) )
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