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Theorem List for Metamath Proof Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcossxp 5101 Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)

Theoremrelrelss 5102 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Theoremunielrel 5103 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)

Theoremrelfld 5104 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Theoremrelresfld 5105 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)

Theoremrelcoi2 5106 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Theoremrelcoi1 5107 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Theoremunidmrn 5108 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)

Theoremrelcnvfld 5109 if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)

Theoremdfdm2 5110 Alternate definition of domain df-dm 4598 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Theoremunixp 5111 The double class union of a non-empty cross product is the union of it members. (Contributed by NM, 17-Sep-2006.)

Theoremunixp0 5112 A cross product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Theoremunixpid 5113 Field of a square cross product. (Contributed by FL, 10-Oct-2009.)

Theoremcnvexg 5114 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)

Theoremcnvex 5115 The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)

Theoremrelcnvexb 5116 A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)

Theoremressn 5117 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremcnviin 5118* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Theoremcnvpo 5119 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcnvso 5120 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)

Theoremcoexg 5121 The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)

Theoremcoex 5122 The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)

Theoremdffun2 5123* Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun3 5124* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun4 5125* Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)

Theoremdffun5 5126* Alternate definition of function. (Contributed by NM, 29-Dec-1996.)

Theoremdffun6f 5127* Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremdffun6 5128* Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.)

Theoremfunmo 5129* A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Theoremfunrel 5130 A function is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfunss 5131 Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)

Theoremfuneq 5132 Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Theoremfuneqi 5133 Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfuneqd 5134 Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Theoremnffun 5135 Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)

Theoremfuneu 5136* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfuneu2 5137* There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)

Theoremdffun7 5138* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun8 5139 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)

Theoremdffun8 5139* Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5138. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremdffun9 5140* Alternate definition of a function. (Contributed by NM, 28-Mar-2007.)

Theoremfunfn 5141 An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.)

Theoremfuni 5142 The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.)

Theoremnfunv 5143 The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)

Theoremfunopg 5144 A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfunopab 5145* A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)

Theoremfunopabeq 5146* A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Theoremfunopab4 5147* A class of ordered pairs of values in the form used by df-mpt 3976 is a function. (Contributed by NM, 17-Feb-2013.)

Theoremfunmpt 5148 A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfunco 5149 The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunres 5150 A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)

Theoremfunssres 5151 The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)

Theoremfun2ssres 5152 Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Theoremfunun 5153 The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Theoremfuncnvsn 5154 The converse singleton of an ordered pair is a function. This is equivalent to funsn 5157 via cnvsn 5061, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.)

Theoremfunsng 5155 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)

Theoremfnsng 5156 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremfunsn 5157 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)

Theoremfunprg 5158 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Theoremfunpr 5159 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Theoremfuntp 5160 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)

Theoremfnsn 5161 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremfnprg 5162 Domain of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfntp 5163 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun0 5164 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)

Theoremfuncnvcnv 5165 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)

Theoremfuncnv2 5166* A simpler equivalence for single-rooted (see funcnv 5167). (Contributed by NM, 9-Aug-2004.)

Theoremfuncnv 5167* The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 5166 for a simpler version. (Contributed by NM, 13-Aug-2004.)

Theoremfuncnv3 5168* A condition showing a class is single-rooted. (See funcnv 5167). (Contributed by NM, 26-May-2006.)

Theoremfun2cnv 5169* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that is not necessarily a function. (Contributed by NM, 13-Aug-2004.)

Theoremsvrelfun 5170 A single-valued relation is a function. (See fun2cnv 5169 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)

Theoremfncnv 5171* Single-rootedness (see funcnv 5167) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)

Theoremfun11 5172* Two ways of stating that is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)

Theoremfununi 5173* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)

Theoremfuncnvuni 5174* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5167 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)

Theoremfun11uni 5175* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)

Theoremfunin 5176 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfunres11 5177 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)

Theoremfuncnvres 5178 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)

Theoremcnvresid 5179 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)

Theoremfuncnvres2 5180 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)

Theoremfunimacnv 5181 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Theoremfunimass1 5182 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)

Theoremfunimass2 5183 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)

Theoremimadif 5184 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)

Theoremimain 5185 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremfunimaexg 5186 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)

Theoremfunimaex 5187 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4028. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)

Theoremisarep1 5188* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by i.e. the class . If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremisarep2 5189* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " i, i, i => o => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5187. (Contributed by NM, 26-Oct-2006.)

Theoremfneq1 5190 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq2 5191 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Theoremfneq1d 5192 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2d 5193 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq12d 5194 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)

Theoremfneq1i 5195 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfneq2i 5196 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)

Theoremnffn 5197 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)

Theoremfnfun 5198 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)

Theoremfnrel 5199 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)

Theoremfndm 5200 The domain of a function. (Contributed by NM, 2-Aug-1994.)

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