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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | imass2 5501 | Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.) |
Theorem | ndmima 5502 | The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998.) (Proof shortened by OpenAI, 3-Jul-2020.) |
Theorem | relcnv 5503 | A converse is a relation. Theorem 12 of [Suppes] p. 62. (Contributed by NM, 29-Oct-1996.) |
Theorem | relbrcnvg 5504 | When is a relation, the sethood assumptions on brcnv 5305 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | eliniseg2 5505 | Eliminate the class existence constraint in eliniseg 5494. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.) |
Theorem | relbrcnv 5506 | When is a relation, the sethood assumptions on brcnv 5305 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | cotrg 5507* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 5508 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 5508. (Revised by Richard Penner, 24-Dec-2019.) |
Theorem | cotr 5508* | Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. Special instance of cotrg 5507. (Contributed by NM, 27-Dec-1996.) |
Theorem | issref 5509* | Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.) |
Theorem | cnvsym 5510* | 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.) |
Theorem | intasym 5511* | 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.) |
Theorem | asymref 5512* | Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5661. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | asymref2 5513* | Two ways of saying a relation is antisymmetric and reflexive. (Contributed by NM, 6-May-2008.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) |
Theorem | intirr 5514* | 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.) |
Theorem | brcodir 5515* | Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.) |
Theorem | codir 5516* | Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Theorem | qfto 5517* | A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Theorem | xpidtr 5518 | A square Cartesian product is a transitive relation. (Contributed by FL, 31-Jul-2009.) |
Theorem | trin2 5519 | The intersection of two transitive classes is transitive. (Contributed by FL, 31-Jul-2009.) |
Theorem | poirr2 5520 | A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) |
Theorem | trinxp 5521 | The relation induced by a transitive relation on a part of its field is transitive. (Taking the intersection of a relation with a square Cartesian product is a way to restrict it to a subset of its field.) (Contributed by FL, 31-Jul-2009.) |
Theorem | soirri 5522 | A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | sotri 5523 | A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | son2lpi 5524 | A strict order relation has no 2-cycle loops. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Theorem | sotri2 5525 | A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.) |
Theorem | sotri3 5526 | A transitivity relation. (Read and implies .) (Contributed by Mario Carneiro, 10-May-2013.) |
Theorem | poleloe 5527 | Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | poltletr 5528 | Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | somin1 5529 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | somincom 5530 | Commutativity of minimum in a total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | somin2 5531 | Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | soltmin 5532 | Being less than a minimum, for a general total order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Theorem | cnvopab 5533* | The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | mptcnv 5534* | The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.) |
Theorem | cnv0 5535 | The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4781, ax-nul 4789, ax-pr 4906. (Revised by KP, 25-Oct-2021.) |
Theorem | cnv0OLD 5536 | Obsolete version of cnv0 5535 as of 25-Oct-2021. (Contributed by NM, 6-Apr-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | cnvi 5537 | 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.) |
Theorem | cnvun 5538 | 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.) |
Theorem | cnvdif 5539 | Distributive law for converse over class difference. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Theorem | cnvin 5540 | 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.) |
Theorem | rnun 5541 | Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.) |
Theorem | rnin 5542 | The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
Theorem | rniun 5543 | The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
Theorem | rnuni 5544* | 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.) |
Theorem | imaundi 5545 | Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
Theorem | imaundir 5546 | The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.) |
Theorem | dminss 5547 | 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.) |
Theorem | imainss 5548 | An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.) |
Theorem | inimass 5549 | The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Theorem | inimasn 5550 | The intersection of the image of singleton. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Theorem | cnvxp 5551 | The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | xp0 5552 | The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
Theorem | xpnz 5553 | The Cartesian product of nonempty classes is nonempty. (Variation of a theorem contributed by Raph Levien, 30-Jun-2006.) (Contributed by NM, 30-Jun-2006.) |
Theorem | xpeq0 5554 | At least one member of an empty Cartesian product is empty. (Contributed by NM, 27-Aug-2006.) |
Theorem | xpdisj1 5555 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Theorem | xpdisj2 5556 | Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Theorem | xpsndisj 5557 | Cartesian products with two different singletons are disjoint. (Contributed by NM, 28-Jul-2004.) |
Theorem | difxp 5558 | Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Theorem | difxp1 5559 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Theorem | difxp2 5560 | Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Theorem | djudisj 5561* | Disjoint unions with disjoint index sets are disjoint. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Theorem | xpdifid 5562* | The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.) |
Theorem | resdisj 5563 | A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | rnxp 5564 | The range of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
Theorem | dmxpss 5565 | The domain of a Cartesian product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.) |
Theorem | rnxpss 5566 | The range of a Cartesian product is a subclass of the second factor. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | rnxpid 5567 | The range of a square Cartesian product. (Contributed by FL, 17-May-2010.) |
Theorem | ssxpb 5568 | A Cartesian product subclass relationship is equivalent to the relationship for it components. (Contributed by NM, 17-Dec-2008.) |
Theorem | xp11 5569 | The Cartesian product of nonempty classes is one-to-one. (Contributed by NM, 31-May-2008.) |
Theorem | xpcan 5570 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
Theorem | xpcan2 5571 | Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
Theorem | ssrnres 5572 | Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.) |
Theorem | rninxp 5573* | Range of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | dminxp 5574* | Domain of the intersection with a Cartesian product. (Contributed by NM, 17-Jan-2006.) |
Theorem | imainrect 5575 | Image of a relation restricted to a rectangular region. (Contributed by Stefan O'Rear, 19-Feb-2015.) |
Theorem | xpima 5576 | The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Theorem | xpima1 5577 | The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Theorem | xpima2 5578 | The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.) |
Theorem | xpimasn 5579 | The image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.) |
Theorem | sossfld 5580 | The base set of a strict order is contained in the field of the relation, except possibly for one element (note that ). (Contributed by Mario Carneiro, 27-Apr-2015.) |
Theorem | sofld 5581 | The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | cnvcnv3 5582* | The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Theorem | dfrel2 5583 | Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.) |
Theorem | dfrel4v 5584* | A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6241 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Theorem | dfrel4 5585* | A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6241 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.) |
Theorem | cnvcnv 5586 | The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) |
Theorem | cnvcnvOLD 5587 | Obsolete proof of cnvcnv 5586 as of 26-Nov-2021. (Contributed by NM, 8-Dec-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | cnvcnv2 5588 | The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) |
Theorem | cnvcnvss 5589 | The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) |
Theorem | cnveqb 5590 | Equality theorem for converse. (Contributed by FL, 19-Sep-2011.) |
Theorem | cnveq0 5591 | A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
Theorem | dfrel3 5592 | Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
Theorem | dmresv 5593 | The domain of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | rnresv 5594 | The range of a universal restriction. (Contributed by NM, 14-May-2008.) |
Theorem | dfrn4 5595 | Range defined in terms of image. (Contributed by NM, 14-May-2008.) |
Theorem | csbrn 5596 | Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | rescnvcnv 5597 | The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | cnvcnvres 5598 | The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007.) |
Theorem | imacnvcnv 5599 | The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) |
Theorem | dmsnn0 5600 | The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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