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Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdmcnvcnv 4501 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 4714). (Contributed by NM, 8-Apr-2007.)

 dom  `' `'  dom
 
Theoremrncnvcnv 4502 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

 ran  `' `'  ran
 
Theoremelreldm 4503 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
 Rel  |^| |^|  dom
 
Theoremrneq 4504 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
 ran  ran
 
Theoremrneqi 4505 Equality inference for range. (Contributed by NM, 4-Mar-2004.)
   =>    
 ran  ran
 
Theoremrneqd 4506 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
   =>     ran  ran
 
Theoremrnss 4507 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
 C_  ran  C_  ran
 
Theorembrelrng 4508 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
 F  G  C  ran  C
 
Theoremopelrng 4509 Membership of second member of an ordered pair in a range. (Contributed by Jim Kingdon, 26-Jan-2019.)
 F  G  <. ,  >.  C  ran  C
 
Theorembrelrn 4510 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
 _V   &     _V   =>     C  ran  C
 
Theoremopelrn 4511 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
 _V   &     _V   =>     <. ,  >.  C  ran  C
 
Theoremreleldm 4512 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
 Rel  R  R  dom  R
 
Theoremrelelrn 4513 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)
 Rel  R  R  ran  R
 
Theoremreleldmb 4514* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)
 Rel  R  dom  R  R
 
Theoremrelelrnb 4515* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)
 Rel  R  ran  R  R
 
Theoremreleldmi 4516 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)

 Rel  R   =>     R  dom  R
 
Theoremrelelrni 4517 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

 Rel  R   =>     R  ran  R
 
Theoremdfrnf 4518* 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/_   &     F/_   =>     ran  {  |  }
 
Theoremelrn2 4519* Membership in a range. (Contributed by NM, 10-Jul-1994.)
 _V   =>     ran  <. ,  >.
 
Theoremelrn 4520* Membership in a range. (Contributed by NM, 2-Apr-2004.)
 _V   =>     ran
 
Theoremnfdm 4521 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   =>     F/_ dom
 
Theoremnfrn 4522 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   =>     F/_ ran
 
Theoremdmiin 4523 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

 dom  |^|_  C_  |^|_  dom
 
Theoremrnopab 4524* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

 ran  { <. ,  >.  |  }  {  | 
 }
 
Theoremrnmpt 4525* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
 F  |->    =>     ran 
 F  {  |  }
 
Theoremelrnmpt 4526* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)
 F  |->    =>     C  V  C  ran  F  C
 
Theoremelrnmpt1s 4527* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)
 F  |->    &     D  C   =>     D  C  V  C  ran  F
 
Theoremelrnmpt1 4528 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 F  |->    =>     V  ran  F
 
Theoremelrnmptg 4529* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
 F  |->    =>     V  C  ran  F  C
 
Theoremelrnmpti 4530* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 F  |->    &     _V   =>     C  ran  F  C
 
Theoremrn0 4531 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 4532 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 C  U_  U. ran  |->
 
Theoremdfiin3g 4533 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 C  |^|_  |^|
 ran  |->
 
Theoremdfiun3 4534 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 _V   =>     U_  U. ran  |->
 
Theoremdfiin3 4535 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
 _V   =>     |^|_  |^| ran  |->
 
Theoremriinint 4536* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 X  V  k  I  S  C_  X  X  i^i  |^|_
 k  I  S  |^| { X }  u.  ran  k  I  |->  S
 
Theoremrelrn0 4537 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)
 Rel  (/)  ran  (/)
 
Theoremdmrnssfld 4538 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)
 dom  u.  ran  C_  U. U.
 
Theoremdmexg 4539 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)
 V  dom  _V
 
Theoremrnexg 4540 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.)
 V  ran  _V
 
Theoremdmex 4541 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)
 _V   =>    
 dom  _V
 
Theoremrnex 4542 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.)
 _V   =>    
 ran  _V
 
Theoremiprc 4543 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  _V
 
Theoremdmcoss 4544 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  o.  C_  dom
 
Theoremrncoss 4545 Range of a composition. (Contributed by NM, 19-Mar-1998.)

 ran  o.  C_  ran
 
Theoremdmcosseq 4546 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 ran  C_  dom  dom  o. 
 dom
 
Theoremdmcoeq 4547 Domain of a composition. (Contributed by NM, 19-Mar-1998.)
 dom  ran  dom  o. 
 dom
 
Theoremrncoeq 4548 Range of a composition. (Contributed by NM, 19-Mar-1998.)
 dom  ran  ran  o. 
 ran
 
Theoremreseq1 4549 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
 |`  C  |`  C
 
Theoremreseq2 4550 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
 C  |`  C  |`
 
Theoremreseq1i 4551 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
   =>     |`  C  |`  C
 
Theoremreseq2i 4552 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>     C  |`  C  |`
 
Theoremreseq12i 4553 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
   &     C  D   =>     |`  C  |`  D
 
Theoremreseq1d 4554 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
   =>     |`  C  |`  C
 
Theoremreseq2d 4555 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>     C  |`  C  |`
 
Theoremreseq12d 4556 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
   &     C  D   =>     |`  C  |`  D
 
Theoremnfres 4557 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)
 F/_   &     F/_   =>     F/_  |`
 
Theoremcsbresg 4558 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)
 V  [_  ]_  |`  C  [_  ]_  |`  [_  ]_ C
 
Theoremres0 4559 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)
 |`  (/)  (/)
 
Theoremopelres 4560 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
 _V   =>     <. ,  >.  C  |`  D  <. ,  >.  C  D
 
Theorembrres 4561 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
 _V   =>     C  |`  D  C  D
 
Theoremopelresg 4562 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)
 V  <. ,  >.  C  |`  D  <. ,  >.  C  D
 
Theorembrresg 4563 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)
 V  C  |`  D  C  D
 
Theoremopres 4564 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.)
 _V   =>     D  <. ,  >.  C  |`  D  <. ,  >.  C
 
Theoremresieq 4565 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
 C  _I  |`  C  C
 
Theoremopelresi 4566  <. ,  >. belongs to a restriction of the identity class iff belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
 V  <. ,  >.  _I  |`
 
Theoremresres 4567 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
 |`  |`  C  |`  i^i  C
 
Theoremresundi 4568 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 |`  u.  C  |`  u.  |`  C
 
Theoremresundir 4569 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
 u.  |`  C  |`  C  u.  |`  C
 
Theoremresindi 4570 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
 |` 
 i^i  C  |` 
 i^i  |`  C
 
Theoremresindir 4571 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
 i^i  |`  C  |`  C 
 i^i  |`  C
 
Theoreminres 4572 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
 i^i  |`  C 
 i^i  |`  C
 
Theoremresiun1 4573* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 U_  |`  C  U_  |`  C
 
Theoremresiun2 4574* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
 C  |`  U_ 
 U_  C  |`
 
Theoremdmres 4575 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

 dom  |`  i^i  dom
 
Theoremssdmres 4576 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)
 C_  dom  dom  |`
 
Theoremdmresexg 4577 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
 V  dom  |`  _V
 
Theoremresss 4578 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
 |`  C_
 
Theoremrescom 4579 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
 |`  |`  C  |`  C  |`
 
Theoremssres 4580 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
 C_  |`  C  C_  |`  C
 
Theoremssres2 4581 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 C_  C  |`  C_  C  |`
 
Theoremrelres 4582 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  |`
 
Theoremresabs1 4583 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
 C_  C  |`  C  |`  |`
 
Theoremresabs2 4584 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
 C_  C  |`  |`  C  |`
 
Theoremresidm 4585 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
 |`  |`  |`
 
Theoremresima 4586 A restriction to an image. (Contributed by NM, 29-Sep-2004.)
 |` 
 "  "
 
Theoremresima2 4587 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)
 C_  C  |`  C
 "  "
 
Theoremxpssres 4588 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
 C  C_  X.  |`  C  C  X.
 
Theoremelres 4589* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)
 |`  C  C  <. ,  >.  <. ,  >.
 
Theoremelsnres 4590* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)
 C  _V   =>     |`  { C }  <. C ,  >. 
 <. C ,  >.
 
Theoremrelssres 4591 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)
 Rel 
 dom  C_  |`
 
Theoremresdm 4592 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
 Rel  |`  dom
 
Theoremresexg 4593 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 V  |`  _V
 
Theoremresex 4594 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)
 _V   =>     |`  _V
 
Theoremresopab 4595* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)
 { <. ,  >.  |  }  |`  { <. ,  >.  |  }
 
Theoremresiexg 4596 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)
 V  _I  |`  _V
 
Theoremiss 4597 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.)
 C_  _I  _I  |`  dom
 
Theoremresopab2 4598* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)
 C_  { <. ,  >.  |  }  |` 
 { <. ,  >.  |  }
 
Theoremresmpt 4599* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
 C_  |->  C  |`  |->  C
 
Theoremresmpt3 4600* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)
 |->  C  |`  i^i  |->  C
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