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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.)

Theoremrelelrni 4602 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

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.)

Theoremelrn2 4604* Membership in a range. (Contributed by NM, 10-Jul-1994.)

Theoremelrn 4605* Membership in a range. (Contributed by NM, 2-Apr-2004.)

Theoremnfdm 4606 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfrn 4607 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdmiin 4608 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Theoremrnopab 4609* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

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.)

Theoremelrnmpt 4611* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremelrnmpt1s 4612* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremelrnmpt1 4613 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmptg 4614* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpti 4615* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

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.)

Theoremdfiun3g 4617 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3g 4618 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3 4619 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3 4620 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremriinint 4621* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremrelrn0 4622 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmrnssfld 4623 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Theoremdmexg 4624 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)

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.)

Theoremdmex 4626 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)

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.)

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.)

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.)

Theoremrncoss 4630 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremdmcosseq 4631 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcoeq 4632 Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremrncoeq 4633 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremreseq1 4634 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Theoremreseq2 4635 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

Theoremreseq1i 4636 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2i 4637 Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12i 4638 Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq1d 4639 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremreseq2d 4640 Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremreseq12d 4641 Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)

Theoremnfres 4642 Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremcsbresg 4643 Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremres0 4644 A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.)

Theoremopelres 4645 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Theorembrres 4646 Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)

Theoremopelresg 4647 Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Theorembrresg 4648 Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.)

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.)

Theoremresieq 4650 A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)

Theoremopelresi 4651 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.)

Theoremresres 4652 The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)

Theoremresundi 4653 Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Theoremresundir 4654 Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)

Theoremresindi 4655 Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)

Theoremresindir 4656 Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)

Theoreminres 4657 Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)

Theoremresiun1 4658* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremresiun2 4659* Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Theoremdmres 4660 The domain of a restriction. Exercise 14 of [TakeutiZaring] p. 25. (Contributed by NM, 1-Aug-1994.)

Theoremssdmres 4661 A domain restricted to a subclass equals the subclass. (Contributed by NM, 2-Mar-1997.)

Theoremdmresexg 4662 The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)

Theoremresss 4663 A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrescom 4664 Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremssres 4665 Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremssres2 4666 Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

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.)

Theoremresabs1 4668 Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)

Theoremresabs2 4669 Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresidm 4670 Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)

Theoremresima 4671 A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Theoremresima2 4672 Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Theoremxpssres 4673 Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremelres 4674* Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremelsnres 4675* Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Theoremrelssres 4676 Simplification law for restriction. (Contributed by NM, 16-Aug-1994.)

Theoremresdm 4677 A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)

Theoremresexg 4678 The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremresex 4679 The restriction of a set is a set. (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremresopab 4680* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Theoremresiexg 4681 The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.)

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.)

Theoremresopab2 4683* Restriction of a class abstraction of ordered pairs. (Contributed by NM, 24-Aug-2007.)

Theoremresmpt 4684* Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Theoremresmpt3 4685* Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Theoremdfres2 4686* Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremopabresid 4687* The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Theoremmptresid 4688* The restricted identity expressed with the "maps to" notation. (Contributed by FL, 25-Apr-2012.)

Theoremdmresi 4689 The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.)

Theoremresid 4690 Any relation restricted to the universe is itself. (Contributed by NM, 16-Mar-2004.)

Theoremimaeq1 4691 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq2 4692 Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Theoremimaeq1i 4693 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq2i 4694 Equality theorem for image. (Contributed by NM, 21-Dec-2008.)

Theoremimaeq1d 4695 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq2d 4696 Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Theoremimaeq12d 4697 Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

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.)

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.)

Theoremelimag 4700* Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

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