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Theorem List for Intuitionistic Logic Explorer - 4601-4700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfres2 4601* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
 R  |`  { <. ,  >.  |  R }
 
Theoremopabresid 4602* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

 { <. ,  >.  |  }  _I  |`
 
Theoremmptresid 4603* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)
 |->  _I  |`
 
Theoremdmresi 4604 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

 dom  _I  |`
 
Theoremresid 4605 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)
 Rel  |`  _V
 
Theoremimaeq1 4606 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 " C  " C
 
Theoremimaeq2 4607 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
 C "  C "
 
Theoremimaeq1i 4608 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
   =>     " C  " C
 
Theoremimaeq2i 4609 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
   =>     C "  C "
 
Theoremimaeq1d 4610 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
   =>     " C  " C
 
Theoremimaeq2d 4611 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)
   =>     C "  C "
 
Theoremimaeq12d 4612 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
   &     C  D   =>     " C  " D
 
Theoremdfima2 4613* 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.)
 "  {  |  }
 
Theoremdfima3 4614* 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.)
 "  {  |  <. ,  >.  }
 
Theoremelimag 4615* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)
 V  " C  C
 
Theoremelima 4616* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)
 _V   =>     " C  C
 
Theoremelima2 4617* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 11-Aug-2004.)
 _V   =>     " C  C
 
Theoremelima3 4618* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)
 _V   =>     " C  C  <. ,  >.
 
Theoremnfima 4619 Bound-variable hypothesis builder for image. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 F/_   &     F/_   =>     F/_
 "
 
Theoremnfimad 4620 Deduction version of bound-variable hypothesis builder nfima 4619. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_   &     F/_   =>     F/_ "
 
Theoremimadmrn 4621 The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)
 " dom  ran
 
Theoremimassrn 4622 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.)
 "  C_  ran
 
Theoremimaexg 4623 The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)
 V  "  _V
 
Theoremimai 4624 Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.)
 _I  "
 
Theoremrnresi 4625 The range of the restricted identity function. (Contributed by NM, 27-Aug-2004.)

 ran  _I  |`
 
Theoremresiima 4626 The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
 C_  _I  |` 
 "
 
Theoremima0 4627 Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
 " (/)  (/)
 
Theorem0ima 4628 Image under the empty relation. (Contributed by FL, 11-Jan-2007.)
 (/) "  (/)
 
Theoremcsbima12g 4629 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.)
 C  [_  ]_ F "  [_  ]_ F " [_  ]_
 
Theoremimadisj 4630 A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.)
 "  (/)  dom 
 i^i  (/)
 
Theoremcnvimass 4631 A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.)
 `' "  C_  dom
 
Theoremcnvimarndm 4632 The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)
 `' " ran  dom
 
Theoremimasng 4633* The image of a singleton. (Contributed by NM, 8-May-2005.)
 R " { }  {  |  R }
 
Theoremelreimasng 4634 Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)
 Rel  R  V  R " { }  R
 
Theoremelimasn 4635 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 _V   &     C  _V   =>     C  " { } 
 <. ,  C >.
 
Theoremelimasng 4636 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
 V  C  W  C  " { }  <. ,  C >.
 
Theoremargs 4637* Two ways to express the class of unique-valued arguments of  F, which is the same as the domain of  F whenever  F is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg  F " for this class (for which we have no separate notation). (Contributed by NM, 8-May-2005.)

 {  |  F " { }  { } }  {  |  F }
 
Theoremeliniseg 4638 Membership in an initial segment. The idiom  `' " { }, meaning  {  |  }, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 C  _V   =>     V  C  `' " { }  C
 
Theoremepini 4639 Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
 _V   =>     `'  _E  " { }
 
Theoreminiseg 4640* An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
 V  `' " { } 
 {  |  }
 
Theoremdfse2 4641* Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
 R Se  i^i  `' R " { }  _V
 
Theoremexse2 4642 Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
 R  V  R Se
 
Theoremimass1 4643 Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
 C_  " C  C_  " C
 
Theoremimass2 4644 Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
 C_  C "  C_  C "
 
Theoremndmima 4645 The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.)
 dom  " { }  (/)
 
Theoremrelcnv 4646 A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.)

 Rel  `'
 
Theoremrelbrcnvg 4647 When  R is a relation, the sethood assumptions on brcnv 4461 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)
 Rel  R  `' R  R
 
Theoremrelbrcnv 4648 When  R is a relation, the sethood assumptions on brcnv 4461 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

 Rel  R   =>     `' R  R
 
Theoremcotr 4649* Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 R  o.  R  C_  R  R  R  R
 
Theoremissref 4650* Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
 _I  |`  C_  R  R
 
Theoremcnvsym 4651* Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 `' R  C_  R  R  R
 
Theoremintasym 4652* Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 R  i^i  `' R  C_  _I  R  R
 
Theoremasymref 4653* Two ways of saying a relation is antisymmetric and reflexive.  U. U. R is the field of a relation by relfld 4789. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 R  i^i  `' R  _I  |`  U. U. R  U. U. R R  R
 
Theoremintirr 4654* Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 R  i^i  _I  (/)  R
 
Theorembrcodir 4655* Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)
 V  W  `' R  o.  R  R  R
 
Theoremcodir 4656* Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
 X.  C_  `' R  o.  R  R  R
 
Theoremqfto 4657* A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.)
 X.  C_  R  u.  `' R  R  R
 
Theoremxpidtr 4658 A square cross product  X. is a transitive relation. (Contributed by FL, 31-Jul-2009.)
 X.  o.  X.  C_  X.
 
Theoremtrin2 4659 The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.)
 R  o.  R  C_  R  S  o.  S  C_  S  R  i^i  S  o.  R  i^i  S  C_  R  i^i  S
 
Theorempoirr2 4660 A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
 R  Po  R  i^i  _I  |`  (/)
 
Theoremtrinxp 4661 The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square cross product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.)
 R  o.  R  C_  R  R  i^i  X.  o.  R  i^i  X.  C_  R  i^i  X.
 
Theoremsoirri 4662 A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 R  Or  S   &     R  C_  S  X.  S   =>     R
 
Theoremsotri 4663 A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 R  Or  S   &     R  C_  S  X.  S   =>     R  R C  R C
 
Theoremson2lpi 4664 A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
 R  Or  S   &     R  C_  S  X.  S   =>     R  R
 
Theoremsotri2 4665 A transitivity relation. (Read B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
 R  Or  S   &     R  C_  S  X.  S   =>     S  R  R C  R C
 
Theoremsotri3 4666 A transitivity relation. (Read A < B and C < B implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.)
 R  Or  S   &     R  C_  S  X.  S   =>     C  S  R  C R  R C
 
Theorempoleloe 4667 Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 V  R  u.  _I  R
 
Theorempoltletr 4668 Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 R  Po  X  X  X  C  X  R  R  u.  _I  C  R C
 
Theoremcnvopab 4669* The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 `' { <. ,  >.  |  }  {
 <. ,  >.  |  }
 
Theoremcnv0 4670 The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
 `' (/)  (/)
 
Theoremcnvi 4671 The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 `'  _I  _I
 
Theoremcnvun 4672 The converse of a union is the union of converses. Theorem 16 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 `'  u.  `'  u.  `'
 
Theoremcnvdif 4673 Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.)
 `'  \  `'  \  `'
 
Theoremcnvin 4674 Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
 `'  i^i  `'  i^i  `'
 
Theoremrnun 4675 Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

 ran  u.  ran  u.  ran
 
Theoremrnin 4676 The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

 ran  i^i  C_  ran 
 i^i  ran
 
Theoremrniun 4677 The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

 ran  U_  U_  ran
 
Theoremrnuni 4678* The range of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 17-Mar-2004.) (Revised by Mario Carneiro, 29-May-2015.)

 ran  U.  U_  ran
 
Theoremimaundi 4679 Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
 "  u.  C 
 "  u.  " C
 
Theoremimaundir 4680 The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
 u.  " C 
 " C  u.  " C
 
Theoremdminss 4681 An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)
 dom  R  i^i  C_  `' R " R "
 
Theoremimainss 4682 An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
 R "  i^i  C_  R "  i^i  `' R "
 
Theoreminimass 4683 The image of an intersection (Contributed by Thierry Arnoux, 16-Dec-2017.)
 i^i  " C  C_  " C  i^i 
 " C
 
Theoreminimasn 4684 The intersection of the image of singleton (Contributed by Thierry Arnoux, 16-Dec-2017.)
 C  V  i^i  " { C }  " { C }  i^i 
 " { C }
 
Theoremcnvxp 4685 The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 `'  X.  X.
 
Theoremxp0 4686 The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
 X.  (/)  (/)
 
Theoremxpmlem 4687* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)
 X.
 
Theoremxpm 4688* The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)
 X.
 
Theoremxpeq0r 4689 A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
 (/)  (/)  X.  (/)
 
Theoremxpdisj1 4690 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
 i^i  (/)  X.  C 
 i^i  X.  D  (/)
 
Theoremxpdisj2 4691 Cross products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.)
 i^i  (/)  C  X. 
 i^i  D  X.  (/)
 
Theoremxpsndisj 4692 Cross products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.)
 =/=  D  X.  { }  i^i  C  X.  { D }  (/)
 
Theoremdjudisj 4693* Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 i^i  (/)  U_  { }  X.  C  i^i  U_  { }  X.  D  (/)
 
Theoremresdisj 4694 A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 i^i  (/)  C  |`  |`  (/)
 
Theoremrnxpm 4695* The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37, with non-empty changed to inhabited. (Contributed by Jim Kingdon, 12-Dec-2018.)
 ran  X.
 
Theoremdmxpss 4696 The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)

 dom  X.  C_
 
Theoremrnxpss 4697 The range of a cross product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

 ran  X.  C_
 
Theoremrnxpid 4698 The range of a square cross product. (Contributed by FL, 17-May-2010.)

 ran  X.
 
Theoremssxpbm 4699* A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)
 X. 
 X.  C_  C  X.  D  C_  C  C_  D
 
Theoremssxp1 4700* Cross product subset cancellation. (Contributed by Jim Kingdon, 14-Dec-2018.)
 C  X.  C  C_  X.  C  C_
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