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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreleldmi 4601 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 4602 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 4603* 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 4604* Membership in a range. (Contributed by NM, 10-Jul-1994.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x <. x ,  A >.  e.  B )
 
Theoremelrn 4605* Membership in a range. (Contributed by NM, 2-Apr-2004.)
 |-  A  e.  _V   =>    |-  ( A  e.  ran 
 B 
 <-> 
 E. x  x B A )
 
Theoremnfdm 4606 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 4607 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 4608 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
 |- 
 dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
 
Theoremrnopab 4609* 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 4610* 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 4611* 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 4612* 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 4613 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 4614* 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 4615* 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 )
 
Theoremrn0 4616 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  (/)  =  (/)
 
Theoremdfiun3g 4617 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 4618 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 4619 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 4620 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 4621* 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 ) ) )
 
Theoremrelrn0 4622 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 |-  ( Rel  A  ->  ( A  =  (/)  <->  ran  A  =  (/) ) )
 
Theoremdmrnssfld 4623 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 4624 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 4625 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 4626 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 4627 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 4628 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. (Contributed by NM, 1-Jan-2007.)
 |- 
 -.  _I  e.  _V
 
Theoremdmcoss 4629 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 4630 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |- 
 ran  ( A  o.  B )  C_  ran  A
 
Theoremdmcosseq 4631 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 4632 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  dom  ( A  o.  B )  = 
 dom  B )
 
Theoremrncoeq 4633 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 |-  ( dom  A  =  ran  B  ->  ran  ( A  o.  B )  = 
 ran  A )
 
Theoremreseq1 4634 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2 4635 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
 |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq1i 4636 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   =>    |-  ( A  |`  C )  =  ( B  |`  C )
 
Theoremreseq2i 4637 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  B   =>    |-  ( C  |`  A )  =  ( C  |`  B )
 
Theoremreseq12i 4638 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  |`  C )  =  ( B  |`  D )
 
Theoremreseq1d 4639 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
 
Theoremreseq2d 4640 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  |`  A )  =  ( C  |`  B ) )
 
Theoremreseq12d 4641 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
 
Theoremnfres 4642 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 4643 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ ( B  |`  C )  =  ( [_ A  /  x ]_ B  |`  [_ A  /  x ]_ C ) )
 
Theoremres0 4644 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |-  ( A  |`  (/) )  =  (/)
 
Theoremopelres 4645 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 4646 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 4647 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 4648 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 4649 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 4650 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 ) )
 
Theoremopelresi 4651  <. 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 4652 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
 
Theoremresundi 4653 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 4654 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
 
Theoremresindi 4655 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |-  ( A  |`  ( B  i^i  C ) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
 
Theoremresindir 4656 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
 
Theoreminres 4657 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
 
Theoremresiun1 4658* 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 4659* 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 4660 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 4661 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 |-  ( A  C_  dom  B  <->  dom  ( B  |`  A )  =  A )
 
Theoremdmresexg 4662 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
 
Theoremresss 4663 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |-  ( A  |`  B ) 
 C_  A
 
Theoremrescom 4664 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  C )  |`  B )
 
Theoremssres 4665 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( A  C_  B  ->  ( A  |`  C ) 
 C_  ( B  |`  C ) )
 
Theoremssres2 4666 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 4667 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 4668 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 4669 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( B  C_  C  ->  ( ( A  |`  B )  |`  C )  =  ( A  |`  B )
 )
 
Theoremresidm 4670 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
 
Theoremresima 4671 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |-  ( ( A  |`  B )
 " B )  =  ( A " B )
 
Theoremresima2 4672 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 |-  ( B  C_  C  ->  ( ( A  |`  C )
 " B )  =  ( A " B ) )
 
Theoremxpssres 4673 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 4674* 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 4675* 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 4676 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 |-  ( ( Rel  A  /\  dom  A  C_  B )  ->  ( A  |`  B )  =  A )
 
Theoremresdm 4677 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 |-  ( Rel  A  ->  ( A  |`  dom  A )  =  A )
 
Theoremresexg 4678 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 4679 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  A  e.  _V   =>    |-  ( A  |`  B )  e.  _V
 
Theoremresopab 4680* 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 4681 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e.  _V )
 
Theoremiss 4682 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 4683* 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 4684* 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 4685* 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 4686* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 |-  ( R  |`  A )  =  { <. x ,  y >.  |  ( x  e.  A  /\  x R y ) }
 
Theoremopabresid 4687* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)
 |- 
 { <. x ,  y >.  |  ( x  e.  A  /\  y  =  x ) }  =  (  _I  |`  A )
 
Theoremmptresid 4688* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
 |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
 
Theoremdmresi 4689 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)
 |- 
 dom  (  _I  |`  A )  =  A
 
Theoremresid 4690 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 |-  ( Rel  A  ->  ( A  |`  _V )  =  A )
 
Theoremimaeq1 4691 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2 4692 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  =  B  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq1i 4693 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( A " C )  =  ( B " C )
 
Theoremimaeq2i 4694 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
 |-  A  =  B   =>    |-  ( C " A )  =  ( C " B )
 
Theoremimaeq1d 4695 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A " C )  =  ( B " C ) )
 
Theoremimaeq2d 4696 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C " A )  =  ( C " B ) )
 
Theoremimaeq12d 4697 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A " C )  =  ( B " D ) )
 
Theoremdfima2 4698* 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 4699* 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 4700* 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 ) )
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