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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresexg 4901 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 4902 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresopab 4903* 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 4904 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5589). (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4905 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 4906* 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 4907* 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 4908* 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 )
 
Theoremdfres2 4909* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 4910* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 4911* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 4912 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 4913 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 4914 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 4915 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 4916 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 4917 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 4918 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 4919 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 4920 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 4921* 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 4922* 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 4923* 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 4924* 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 4925* 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 4926* 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 4927 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 4928 Deduction version of bound-variable hypothesis builder nfima 4927. (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 ) )
 
Theoremcsbima12g 4929 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 ) )
 
Theoremcsbima12gALT 4930 Move class substitution in and out of the image of a function. (This is csbima12g 4929 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbima12gALTVD 27363. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F "
 [_ A  /  x ]_ B ) )
 
Theoremimadmrn 4931 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 4932 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 4933 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 )
 
Theoremimai 4934 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 4935 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 4936 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 4937 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 4938 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremimadisj 4939 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 4940 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 4941 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 4942* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremrelimasn 4943* The image of a singleton. (Contributed by NM, 20-May-1998.)
 |-  ( Rel  R  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelrelimasn 4944 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( Rel  R  ->  ( B  e.  ( R
 " { A }
 ) 
 <->  A R B ) )
 
Theoremelimasn 4945 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 4946 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 ) )
 
Theoremelimasni 4947 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
 |-  ( C  e.  ( A " { B }
 )  ->  B A C )
 
Theoremargs 4948* 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). Observe the resemblance to our df-fv 4608, which was based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
 |- 
 { x  |  E. y ( F " { x } )  =  { y } }  =  { x  |  E! y  x F y }
 
Theoremeliniseg 4949 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 4950 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 4951* 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 } )
 
Theoremdffr3 4952* Alternate definition of well-founded relation. Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)
 |-  ( R  Fr  A  <->  A. x ( ( x 
 C_  A  /\  x  =/= 
 (/) )  ->  E. y  e.  x  ( x  i^i  ( `' R " { y } )
 )  =  (/) ) )
 
Theoremdfse2 4953* 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 4954 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 4955 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 4956 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 4957 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
 
Theoremrelcnv 4958 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)
 |- 
 Rel  `' A
 
Theoremrelbrcnvg 4959 When  R is a relation, the sethood assumptions on brcnv 4771 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( Rel  R  ->  ( A `' R B  <->  B R A ) )
 
Theoremeliniseg2 4960 Eliminate the class existence constraint in eliniseg 4949. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.)
 |-  ( Rel  A  ->  ( C  e.  ( `' A " { B } )  <->  C A B ) )
 
Theoremrelbrcnv 4961 When  R is a relation, the sethood assumptions on brcnv 4771 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcotr 4962* 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 4963* 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 4964* 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 4965* 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 4966* Two ways of saying a relation is antisymmetric and reflexive.  U. U. R is the field of a relation by relfld 5104. (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
 ) )
 
Theoremasymref2 4967* 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  i^i  `' R )  =  (  _I  |`  U. U. R ) 
 <->  ( A. x  e. 
 U. U. R x R x  /\  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y ) ) )
 
Theoremintirr 4968* 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 4969* 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 4970* 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 4971* 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 4972 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 4973 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 4974 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
 |-  ( R  Po  A  ->  ( R  i^i  (  _I  |`  A ) )  =  (/) )
 
Theoremtrinxp 4975 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 4976 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 4977 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 4978 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 4979 A transitivity relation. (Read  A  <_  B 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 4980 A transitivity relation. (Read  A  <  B and  B  <_  C 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 )
 
TheoremsoirriOLD 4981 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   =>    |-  -.  A R A
 
TheoremsotriOLD 4982 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  (
 ( A R B  /\  B R C ) 
 ->  A R C )
 
Theoremson2lpiOLD 4983 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  R  Or  S   &    |-  R  C_  ( S  X.  S )   &    |-  B  e.  _V   =>    |- 
 -.  ( A R B  /\  B R A )
 
Theorempoleloe 4984 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 4985 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 ) )
 
Theoremsomin1 4986 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B ) ( R  u.  _I  ) A )
 
Theoremsomincom 4987 Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B )  =  if ( B R A ,  B ,  A )
 )
 
Theoremsomin2 4988 Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X ) )  ->  if ( A R B ,  A ,  B ) ( R  u.  _I  ) B )
 
Theoremsoltmin 4989 Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 |-  ( ( R  Or  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  ( A R if ( B R C ,  B ,  C )  <->  ( A R B  /\  A R C ) ) )
 
Theoremcnvopab 4990* 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 }
 
Theoremcnv0 4991 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 |-  `' (/)  =  (/)
 
Theoremcnvi 4992 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 4993 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 4994 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 |-  `' ( A  \  B )  =  ( `' A  \  `' B )
 
Theoremcnvin 4995 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 4996 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 4997 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 4998 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 4999* 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 5000 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 ) )
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