HomeHome Metamath Proof Explorer
Theorem List (p. 50 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelrnmpt1s 4901* 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 4902 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 4903* 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 4904* 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 4905 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 4906 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 4907 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 4908 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 4909* 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 4910 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 4911 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
 
Theoremdmrnssfld 4912 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 4913 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 4914 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 4915 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 4916 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 4917 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 16935. (Contributed by NM, 1-Jan-2007.)
 |- 
 -.  _I  e.  _V
 
Theoremdmcoss 4918 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 4919 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |- 
 ran  (  A  o.  B )  C_  ran  A
 
Theoremdmcosseq 4920 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 4921 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  dom  (  A  o.  B )  = 
 dom  B )
 
Theoremrncoeq 4922 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  ran  (  A  o.  B )  = 
 ran  A )
 
Theoremreseq1 4923 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2 4924 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
 |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq1i 4925 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   =>    |-  ( A  |`  C )  =  ( B  |`  C )
 
Theoremreseq2i 4926 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( C  |`  A )  =  ( C  |`  B )
 
Theoremreseq12i 4927 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  |`  C )  =  ( B  |`  D )
 
Theoremreseq1d 4928 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2d 4929 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 4930 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 4931 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 4932 Distribute proper substitution through the restriction of a class. csbresg 4932 is derived from the virtual deduction proof csbresgVD 27705. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 4933 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 4934 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 4935 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 4936 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 4937 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 4938 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 4939 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 4940  <. 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 4941  <. 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 4942 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 4943 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 4944 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 4945 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 4946 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 4947 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 4948* 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 4949* 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 4950 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 4951 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 |-  ( A  C_  dom  B  <->  dom  (  B  |`  A )  =  A )
 
Theoremdmresexg 4952 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 |-  ( B  e.  V  ->  dom  (  A  |`  B )  e.  _V )
 
Theoremresss 4953 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B ) 
 C_  A
 
Theoremrescom 4954 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
 
Theoremssres 4955 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  C_  B  ->  ( A  |`  C ) 
 C_  ( B  |`  C ) )
 
Theoremssres2 4956 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 4957 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 4958 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 4959 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( B  C_  C  ->  ( ( A  |`  B )  |`  C )  =  ( A  |`  B )
 )
 
Theoremresidm 4960 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
 
Theoremresima 4961 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |-  ( ( A  |`  B )
 " B )  =  ( A " B )
 
Theoremresima2 4962 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )
 " B )  =  ( A " B ) )
 
Theoremxpssres 4963 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 4964* 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 4965* 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 4966 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 4967 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 4968 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 4969 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresopab 4970* 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 4971 The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 5657). (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4972 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 4973* 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 4974* 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 4975* 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 4976* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 4977* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 4978* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 4979 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 4980 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 4981 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 4982 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 4983 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 4984 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 4985 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 4986 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 4987 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 4988* 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 4989* 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 4990* 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 4991* 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 4992* 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 4993* 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 4994 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 4995 Deduction version of bound-variable hypothesis builder nfima 4994. (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 4996 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 4997 Move class substitution in and out of the image of a function. (This is csbima12g 4996 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbima12gALTVD 27707. 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 4998 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 4999 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 5000 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
  Copyright terms: Public domain < Previous  Next >