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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdmcnvcnv 4901 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5124). (Contributed by NM, 8-Apr-2007.)
 |- 
 dom  `' `' A  =  dom  A
 
Theoremrncnvcnv 4902 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |- 
 ran  `' `' A  =  ran  A
 
Theoremelreldm 4903 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
 |-  ( ( Rel  A  /\  B  e.  A ) 
 ->  |^| |^| B  e.  dom  A )
 
Theoremrneq 4904 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ran  A  =  ran  B )
 
Theoremrneqi 4905 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  ran  A  =  ran  B
 
Theoremrneqd 4906 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ran  A  =  ran  B )
 
Theoremrnss 4907 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ran  A  C_  ran  B )
 
Theorembrelrng 4908 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
 |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B ) 
 ->  B  e.  ran  C )
 
Theorembrelrn 4909 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A C B  ->  B  e.  ran  C )
 
Theoremopelrn 4910 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  B  e.  ran  C )
 
Theoremreleldm 4911 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  A  e.  dom  R )
 
Theoremrelelrn 4912 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
 |-  ( ( Rel  R  /\  A R B ) 
 ->  B  e.  ran  R )
 
Theoremreleldmb 4913* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
 
Theoremrelelrnb 4914* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
 
Theoremreleldmi 4915 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  A  e.  dom  R )
 
Theoremrelelrni 4916 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)
 |- 
 Rel  R   =>    |-  ( A R B  ->  B  e.  ran  R )
 
Theoremdfrnf 4917* Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  ran  A  =  { y  |  E. x  x A y }
 
Theoremelrn2 4918* Membership in a range. (Contributed by NM, 10-Jul-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x <. x ,  A >.  e.  B )
 
Theoremelrn 4919* Membership in a range. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x  x B A )
 
Theoremnfdm 4920 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x dom  A
 
Theoremnfrn 4921 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ran  A
 
Theoremdmiin 4922 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
 |- 
 dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
 
Theoremcsbrng 4923 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 ran  B  =  ran  [_ A  /  x ]_ B )
 
Theoremrnopab 4924* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |- 
 ran  { <. x ,  y >.  |  ph }  =  { y  |  E. x ph
 }
 
Theoremrnmpt 4925* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ran 
 F  =  { y  |  E. x  e.  A  y  =  B }
 
Theoremelrnmpt 4926* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 )
 
Theoremelrnmpt1s 4927* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  =  D  ->  B  =  C )   =>    |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F )
 
Theoremelrnmpt1 4928 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ( x  e.  A  /\  B  e.  V )  ->  B  e.  ran 
 F )
 
Theoremelrnmptg 4929* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B ) )
 
Theoremelrnmpti 4930* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  B  e.  _V   =>    |-  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 
Theoremdfiun3g 4931 Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
 
Theoremdfiin3g 4932 Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A. x  e.  A  B  e.  C  -> 
 |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
 
Theoremdfiun3 4933 Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )
 
Theoremdfiin3 4934 Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   =>    |-  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B )
 
Theoremriinint 4935* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( X  e.  V  /\  A. k  e.  I  S  C_  X )  ->  ( X  i^i  |^|_
 k  e.  I  S )  =  |^| ( { X }  u.  ran  (
 k  e.  I  |->  S ) ) )
 
Theoremrn0 4936 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)
 |- 
 ran  (/)  =  (/)
 
Theoremrelrn0 4937 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
 
Theoremdmrnssfld 4938 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
 |-  ( dom  A  u.  ran 
 A )  C_  U. U. A
 
Theoremdmexg 4939 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  dom  A  e.  _V )
 
Theoremrnexg 4940 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.)
 |-  ( A  e.  V  ->  ran  A  e.  _V )
 
Theoremdmex 4941 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
 |-  A  e.  _V   =>    |-  dom  A  e.  _V
 
Theoremrnex 4942 The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
 |-  A  e.  _V   =>    |-  ran  A  e.  _V
 
Theoremiprc 4943 The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set, as in idcn 16987. (Contributed by NM, 1-Jan-2007.)
 |- 
 -.  _I  e.  _V
 
Theoremdmcoss 4944 Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  ( A  o.  B )  C_  dom  B
 
Theoremrncoss 4945 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |- 
 ran  ( A  o.  B )  C_  ran  A
 
Theoremdmcosseq 4946 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ran  B  C_  dom 
 A  ->  dom  ( A  o.  B )  = 
 dom  B )
 
Theoremdmcoeq 4947 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  dom  ( A  o.  B )  = 
 dom  B )
 
Theoremrncoeq 4948 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  ran  ( A  o.  B )  = 
 ran  A )
 
Theoremreseq1 4949 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2 4950 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
 |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq1i 4951 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   =>    |-  ( A  |`  C )  =  ( B  |`  C )
 
Theoremreseq2i 4952 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( C  |`  A )  =  ( C  |`  B )
 
Theoremreseq12i 4953 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  |`  C )  =  ( B  |`  D )
 
Theoremreseq1d 4954 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2d 4955 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 4956 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 4957 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 4958 Distribute proper substitution through the restriction of a class. csbresg 4958 is derived from the virtual deduction proof csbresgVD 28671. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 4959 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 4960 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 4961 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 4962 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 4963 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 4964 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 4965 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 4966  <. 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 4967  <. 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 4968 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 4969 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 4970 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 4971 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 4972 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 4973 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 4974* 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 4975* 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 4976 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 4977 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 |-  ( A  C_  dom  B  <->  dom  ( B  |`  A )  =  A )
 
Theoremdmresexg 4978 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
 
Theoremresss 4979 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B ) 
 C_  A
 
Theoremrescom 4980 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
 
Theoremssres 4981 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  C_  B  ->  ( A  |`  C ) 
 C_  ( B  |`  C ) )
 
Theoremssres2 4982 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 4983 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 4984 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 4985 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( B  C_  C  ->  ( ( A  |`  B )  |`  C )  =  ( A  |`  B )
 )
 
Theoremresidm 4986 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
 
Theoremresima 4987 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |-  ( ( A  |`  B )
 " B )  =  ( A " B )
 
Theoremresima2 4988 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )
 " B )  =  ( A " B ) )
 
Theoremxpssres 4989 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 4990* 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 4991* 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 4992 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 4993 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 4994 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 4995 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresopab 4996* 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 4997 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5737). (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4998 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 4999* 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 5000* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
 |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
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