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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreseq1i 5101 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   =>    |-  ( A  |`  C )  =  ( B  |`  C )
 
Theoremreseq2i 5102 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( C  |`  A )  =  ( C  |`  B )
 
Theoremreseq12i 5103 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  |`  C )  =  ( B  |`  D )
 
Theoremreseq1d 5104 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2d 5105 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 5106 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 5107 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  |`  B )
 
Theoremcsbresg 5108 Distribute proper substitution through the restriction of a class. csbresg 5108 is derived from the virtual deduction proof csbresgVD 28716. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 5109 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 5110 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
 |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  |`  D )  <->  ( <. A ,  B >.  e.  C  /\  A  e.  D )
 )
 
Theorembrres 5111 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
 |-  B  e.  _V   =>    |-  ( A ( C  |`  D ) B 
 <->  ( A C B  /\  A  e.  D ) )
 
Theoremopelresg 5112 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
 |-  ( B  e.  V  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
 ( <. A ,  B >.  e.  C  /\  A  e.  D ) ) )
 
Theorembrresg 5113 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
 |-  ( B  e.  V  ->  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) ) )
 
Theoremopres 5114 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  e.  _V   =>    |-  ( A  e.  D  ->  ( <. A ,  B >.  e.  ( C  |`  D )  <->  <. A ,  B >.  e.  C ) )
 
Theoremresieq 5115 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
 |-  ( ( B  e.  A  /\  C  e.  A )  ->  ( B (  _I  |`  A ) C 
 <->  B  =  C ) )
 
TheoremopelresiOLD 5116  <. A ,  A >. belongs to a restriction of the identity class iff  A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  e.  V  ->  ( A  e.  B  <->  <. A ,  A >.  e.  (  _I  |`  B ) ) )
 
Theoremopelresi 5117  <. 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.)
 |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <->  A  e.  B )
 )
 
Theoremresres 5118 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 5119 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |-  ( A  |`  ( B  u.  C ) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
 
Theoremresundir 5120 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 5121 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 5122 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 5123 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 5124* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
 
Theoremresiun2 5125* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 |-  ( C  |`  U_ x  e.  A  B )  = 
 U_ x  e.  A  ( C  |`  B )
 
Theoremdmres 5126 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)
 |- 
 dom  ( A  |`  B )  =  ( B  i^i  dom 
 A )
 
Theoremssdmres 5127 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 |-  ( A  C_  dom  B  <->  dom  ( B  |`  A )  =  A )
 
Theoremdmresexg 5128 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
 
Theoremresss 5129 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B ) 
 C_  A
 
Theoremrescom 5130 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
 
Theoremssres 5131 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  C_  B  ->  ( A  |`  C ) 
 C_  ( B  |`  C ) )
 
Theoremssres2 5132 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  C_  B  ->  ( C  |`  A ) 
 C_  ( C  |`  B ) )
 
Theoremrelres 5133 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  ( A  |`  B )
 
Theoremresabs1 5134 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )  |`  B )  =  ( A  |`  B )
 )
 
Theoremresabs2 5135 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( B  C_  C  ->  ( ( A  |`  B )  |`  C )  =  ( A  |`  B )
 )
 
Theoremresidm 5136 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
 
Theoremresima 5137 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |-  ( ( A  |`  B )
 " B )  =  ( A " B )
 
Theoremresima2 5138 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )
 " B )  =  ( A " B ) )
 
Theoremxpssres 5139 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 5140* 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 5141* 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 5142 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 5143 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 5144 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 5145 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresopab 5146* 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 5147 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5916). (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 5148 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 5149* 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 5150* 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 5151* 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 5152* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 5153* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 5154* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 5155 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 5156 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 5157 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 5158 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 5159 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 5160 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 5161 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 5162 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 5163 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 5164* 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 5165* 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 5166* 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 5167* 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 5168* 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 5169* 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 5170 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 5171 Deduction version of bound-variable hypothesis builder nfima 5170. (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 5172 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 5173 Move class substitution in and out of the image of a function. (This is csbima12g 5172 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbima12gALTVD 28718. 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.) (New usage is discouraged.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F "
 [_ A  /  x ]_ B ) )
 
Theoremimadmrn 5174 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 5175 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 5176 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 5177 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 |-  (  _I  " A )  =  A
 
Theoremrnresi 5178 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 ran  (  _I  |`  A )  =  A
 
Theoremresiima 5179 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 |-  ( B  C_  A  ->  ( (  _I  |`  A )
 " B )  =  B )
 
Theoremima0 5180 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 |-  ( A " (/) )  =  (/)
 
Theorem0ima 5181 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 |-  ( (/) " A )  =  (/)
 
Theoremimadisj 5182 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 5183 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 5184 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 5185* The image of a singleton. (Contributed by NM, 8-May-2005.)
 |-  ( A  e.  B  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremrelimasn 5186* The image of a singleton. (Contributed by NM, 20-May-1998.)
 |-  ( Rel  R  ->  ( R " { A } )  =  {
 y  |  A R y } )
 
Theoremelrelimasn 5187 Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)
 |-  ( Rel  R  ->  ( B  e.  ( R
 " { A }
 ) 
 <->  A R B ) )
 
Theoremelimasn 5188 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 5189 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 5190 Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)
 |-  ( C  e.  ( A " { B }
 )  ->  B A C )
 
Theoremargs 5191* 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 the alternative definition dffv4 5684 of function value, which is 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 5192 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 5193 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 5194* 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 5195* 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 5196* 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 5197 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 |-  ( R  e.  V  ->  R Se  A )
 
Theoremimass1 5198 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 |-  ( A  C_  B  ->  ( A " C )  C_  ( B " C ) )
 
Theoremimass2 5199 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 5200 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 |-  ( -.  A  e.  dom 
 B  ->  ( B " { A } )  =  (/) )
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