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Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqbrrdva 5001* Deduction from extensionality principle for relations, given an equivalence only on the relation's domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)

Theorembrco 5002* Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremopelco 5003* Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theoremcnvss 5004 Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Theoremcnveq 5005 Equality theorem for converse. (Contributed by NM, 13-Aug-1995.)

Theoremcnveqi 5006 Equality inference for converse. (Contributed by NM, 23-Dec-2008.)

Theoremcnveqd 5007 Equality deduction for converse. (Contributed by NM, 6-Dec-2013.)

Theoremelcnv 5008* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)

Theoremelcnv2 5009* Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)

Theoremnfcnv 5010 Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremopelcnvg 5011 Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theorembrcnvg 5012 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)

Theoremopelcnv 5013 Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.)

Theorembrcnv 5014 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)

Theoremcnvco 5015 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremcnvuni 5016* The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)

Theoremdfdm3 5017* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremdfrn2 5018* Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Theoremdfrn3 5019* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Theoremelrn2g 5020* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremelrng 5021* Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdfdm4 5022 Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)

Theoremdfdmf 5023* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremeldmg 5024* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremeldm2g 5025* Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremeldm 5026* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)

Theoremeldm2 5027* Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)

Theoremdmss 5028 Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeq 5029 Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)

Theoremdmeqi 5030 Equality inference for domain. (Contributed by NM, 4-Mar-2004.)

Theoremdmeqd 5031 Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)

Theoremopeldm 5032 Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldm 5033 Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Theorembreldmg 5034 Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Theoremdmun 5035 The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmin 5036 The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)

Theoremdmiun 5037 The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)

Theoremdmuni 5038* The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)

Theoremdmopab 5039* The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremdmopabss 5040* Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdmopab3 5041* The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)

Theoremdm0 5042 The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmi 5043 The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmv 5044 The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)

Theoremdm0rn0 5045 An empty domain implies an empty range. (Contributed by NM, 21-May-1998.)

Theoremreldm0 5046 A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmxp 5047 The domain of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmxpid 5048 The domain of a square cross product. (Contributed by NM, 28-Jul-1995.)

Theoremdmxpin 5049 The domain of the intersection of two square cross products. Unlike dmin 5036, equality holds. (Contributed by NM, 29-Jan-2008.)

Theoremxpid11 5050 The cross product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcnvcnv 5051 The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5280). (Contributed by NM, 8-Apr-2007.)

Theoremrncnvcnv 5052 The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Theoremelreldm 5053 The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)

Theoremrneq 5054 Equality theorem for range. (Contributed by NM, 29-Dec-1996.)

Theoremrneqi 5055 Equality inference for range. (Contributed by NM, 4-Mar-2004.)

Theoremrneqd 5056 Equality deduction for range. (Contributed by NM, 4-Mar-2004.)

Theoremrnss 5057 Subset theorem for range. (Contributed by NM, 22-Mar-1998.)

Theorembrelrng 5058 The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)

Theorembrelrn 5059 The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)

Theoremopelrn 5060 Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)

Theoremreleldm 5061 The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)

Theoremrelelrn 5062 The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)

Theoremreleldmb 5063* Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)

Theoremrelelrnb 5064* Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)

Theoremreleldmi 5065 The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.)

Theoremrelelrni 5066 The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.)

Theoremdfrnf 5067* 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 5068* Membership in a range. (Contributed by NM, 10-Jul-1994.)

Theoremelrn 5069* Membership in a range. (Contributed by NM, 2-Apr-2004.)

Theoremnfdm 5070 Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfrn 5071 Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdmiin 5072 Domain of an intersection. (Contributed by FL, 15-Oct-2012.)

Theoremcsbrng 5073 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremrnopab 5074* The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremrnmpt 5075* The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpt 5076* The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.)

Theoremelrnmpt1s 5077* Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Theoremelrnmpt1 5078 Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmptg 5079* Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremelrnmpti 5080* Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3g 5081 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3g 5082 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiun3 5083 Alternate definition of indexed union when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremdfiin3 5084 Alternate definition of indexed intersection when is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremriinint 5085* Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremrn0 5086 The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.)

Theoremrelrn0 5087 A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Theoremdmrnssfld 5088 The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Theoremdmexg 5089 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.)

Theoremrnexg 5090 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 5091 The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Jul-2008.)

Theoremrnex 5092 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 5093 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, as in idcn 17275. (Contributed by NM, 1-Jan-2007.)

Theoremdmcoss 5094 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 5095 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremdmcosseq 5096 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremdmcoeq 5097 Domain of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremrncoeq 5098 Range of a composition. (Contributed by NM, 19-Mar-1998.)

Theoremreseq1 5099 Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Theoremreseq2 5100 Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)

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