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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoeq12d 4801 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  o.  C )  =  ( B  o.  D ) )
 
Theoremnfco 4802 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  o.  B )
 
Theorembrcog 4803* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) ) )
 
Theoremopelco2g 4804* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( <. A ,  x >.  e.  D  /\  <. x ,  B >.  e.  C ) ) )
 
Theorembrco 4805* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A ( C  o.  D ) B  <->  E. x ( A D x  /\  x C B ) )
 
Theoremopelco 4806* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  ( C  o.  D )  <->  E. x ( A D x  /\  x C B ) )
 
Theoremcnvss 4807 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  `' A  C_  `' B )
 
Theoremcnveq 4808 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)
 |-  ( A  =  B  ->  `' A  =  `' B )
 
Theoremcnveqi 4809 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)
 |-  A  =  B   =>    |-  `' A  =  `' B
 
Theoremcnveqd 4810 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  `' A  =  `' B )
 
Theoremelcnv 4811* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  y R x ) )
 
Theoremelcnv2 4812* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
 |-  ( A  e.  `' R 
 <-> 
 E. x E. y
 ( A  =  <. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
 
Theoremnfcnv 4813 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x `' A
 
Theoremopelcnvg 4814 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
 
Theorembrcnvg 4815 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
 |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B  <->  B R A ) )
 
Theoremopelcnv 4816 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R )
 
Theorembrcnv 4817 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A `' R B 
 <->  B R A )
 
Theoremcnvco 4818 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
 
Theoremcnvuni 4819* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
 |-  `' U. A  =  U_ x  e.  A  `' x
 
Theoremdfdm3 4820* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  { x  |  E. y <. x ,  y >.  e.  A }
 
Theoremdfrn2 4821* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x  x A y }
 
Theoremdfrn3 4822* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
 |- 
 ran  A  =  {
 y  |  E. x <. x ,  y >.  e.  A }
 
Theoremelrn2g 4823* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x <. x ,  A >.  e.  B ) )
 
Theoremelrng 4824* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
 |-  ( A  e.  V  ->  ( A  e.  ran  B  <->  E. x  x B A ) )
 
Theoremdfdm4 4825 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
 |- 
 dom  A  =  ran  `'  A
 
Theoremdfdmf 4826* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ y A   =>    |-  dom  A  =  { x  |  E. y  x A y }
 
Theoremeldmg 4827* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y  A B y ) )
 
Theoremeldm2g 4828* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
 |-  ( A  e.  V  ->  ( A  e.  dom  B  <->  E. y <. A ,  y >.  e.  B ) )
 
Theoremeldm 4829* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  dom 
 B 
 <-> 
 E. y  A B y )
 
Theoremeldm2 4830* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  dom 
 B 
 <-> 
 E. y <. A ,  y >.  e.  B )
 
Theoremdmss 4831 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  C_  B  ->  dom  A  C_  dom  B )
 
Theoremdmeq 4832 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( A  =  B  ->  dom  A  =  dom  B )
 
Theoremdmeqi 4833 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  dom  A  =  dom  B
 
Theoremdmeqd 4834 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  dom  A  =  dom  B )
 
Theoremopeldm 4835 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  A  e.  dom  C )
 
Theorembreldm 4836 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A R B  ->  A  e.  dom  R )
 
Theorembreldmg 4837 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B ) 
 ->  A  e.  dom  R )
 
Theoremdmun 4838 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  (  A  u.  B )  =  ( dom  A  u.  dom  B )
 
Theoremdmin 4839 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
 |- 
 dom  (  A  i^i  B )  C_  ( dom  A  i^i  dom  B )
 
Theoremdmiun 4840 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |- 
 dom  U_  x  e.  A  B  =  U_ x  e.  A  dom  B
 
Theoremdmuni 4841* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
 |- 
 dom  U.  A  =  U_ x  e.  A  dom  x
 
Theoremdmopab 4842* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
 |- 
 dom  { <. x ,  y >.  |  ph }  =  { x  |  E. y ph }
 
Theoremdmopabss 4843* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
 |- 
 dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremdmopab3 4844* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
 |-  ( A. x  e.  A  E. y ph  <->  dom  {
 <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
 
Theoremdm0 4845 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  (/)  =  (/)
 
Theoremdmi 4846 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |- 
 dom  _I  =  _V
 
Theoremdmv 4847 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
 |- 
 dom  _V  =  _V
 
Theoremdm0rn0 4848 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)
 |-  ( dom  A  =  (/)  <->  ran 
 A  =  (/) )
 
Theoremreldm0 4849 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  dom  A  =  (/) ) )
 
Theoremdmxp 4850 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )
 
Theoremdmxpid 4851 The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)
 |- 
 dom  (  A  X.  A )  =  A
 
Theoremdmxpin 4852 The domain of the intersection of two square cross products. Unlike dmin 4839, equality holds. (Contributed by NM, 29-Jan-2008.)
 |- 
 dom  ( ( A  X.  A )  i^i  ( B  X.  B ) )  =  ( A  i^i  B )
 
Theoremxpid11 4853 The cross product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  X.  A )  =  ( B  X.  B )  <->  A  =  B )
 
Theoremdmcnvcnv 4854 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5077). (Contributed by NM, 8-Apr-2007.)
 |- 
 dom  `' `' A  =  dom  A
 
Theoremrncnvcnv 4855 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
 |- 
 ran  `' `' A  =  ran  A
 
Theoremelreldm 4856 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 4857 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ran  A  =  ran  B )
 
Theoremrneqi 4858 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
 |-  A  =  B   =>    |-  ran  A  =  ran  B
 
Theoremrneqd 4859 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ran  A  =  ran  B )
 
Theoremrnss 4860 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ran  A  C_  ran  B )
 
Theorembrelrng 4861 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 4862 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 4863 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 4864 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 4865 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 4866* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  dom  R  <->  E. x  A R x ) )
 
Theoremrelelrnb 4867* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
 |-  ( Rel  R  ->  ( A  e.  ran  R  <->  E. x  x R A ) )
 
Theoremreleldmi 4868 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 4869 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 4870* 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 4871* Membership in a range. (Contributed by NM, 10-Jul-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x <. x ,  A >.  e.  B )
 
Theoremelrn 4872* Membership in a range. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x  x B A )
 
Theoremnfdm 4873 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 4874 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 4875 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
 |- 
 dom  |^|_  x  e.  A  B  C_  |^|_ x  e.  A  dom  B
 
Theoremcsbrng 4876 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 4877* 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 4878* 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 4879* 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 4880* 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 4881 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 4882* 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 4883* 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 4884 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 4885 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 4886 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 4887 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 4888* 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 4889 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 4890 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
 
Theoremdmrnssfld 4891 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 4892 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 4893 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 4894 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 4895 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 4896 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 16914. (Contributed by NM, 1-Jan-2007.)
 |- 
 -.  _I  e.  _V
 
Theoremdmcoss 4897 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 4898 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |- 
 ran  (  A  o.  B )  C_  ran  A
 
Theoremdmcosseq 4899 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 4900 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  dom  (  A  o.  B )  = 
 dom  B )
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