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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cores 5101 | Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | resco 5102 | Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.) |
Theorem | imaco 5103 | Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) |
Theorem | rnco 5104 | The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) |
Theorem | rnco2 5105 | The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.) |
Theorem | dmco 5106 | The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.) |
Theorem | coiun 5107* | Composition with an indexed union. (Contributed by NM, 21-Dec-2008.) |
Theorem | cocnvcnv1 5108 | A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cocnvcnv2 5109 | A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.) |
Theorem | cores2 5110 | Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.) |
Theorem | co02 5111 | Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Theorem | co01 5112 | Composition with the empty set. (Contributed by NM, 24-Apr-2004.) |
Theorem | coi1 5113 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coi2 5114 | Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Theorem | coires1 5115 | Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Theorem | coass 5116 | Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.) |
Theorem | relcnvtr 5117 | A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) |
Theorem | relssdmrn 5118 | A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) |
Theorem | cnvssrndm 5119 | The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Theorem | cossxp 5120 | Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Theorem | cossxp2 5121 | The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvres 5122 | Restricting a relation and a converse relation when they are composed together. (Contributed by BJ, 10-Jul-2022.) |
Theorem | cocnvss 5123 | Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.) |
Theorem | relrelss 5124 | Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.) |
Theorem | unielrel 5125 | The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.) |
Theorem | relfld 5126 | The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.) |
Theorem | relresfld 5127 | Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) |
Theorem | relcoi2 5128 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Theorem | relcoi1 5129 | Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) |
Theorem | unidmrn 5130 | The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Theorem | relcnvfld 5131 | if is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Theorem | dfdm2 5132 | Alternate definition of domain df-dm 4608 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Theorem | unixpm 5133* | The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | unixp0im 5134 | The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvexg 5135 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.) |
Theorem | cnvex 5136 | The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Theorem | relcnvexb 5137 | A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.) |
Theorem | ressn 5138 | Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | cnviinm 5139* | The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Theorem | cnvpom 5140* | The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Theorem | cnvsom 5141* | The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Theorem | coexg 5142 | The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.) |
Theorem | coex 5143 | The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.) |
Theorem | xpcom 5144* | Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.) |
Syntax | cio 5145 | Extend class notation with Russell's definition description binder (inverted iota). |
Theorem | iotajust 5146* | Soundness justification theorem for df-iota 5147. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Definition | df-iota 5147* |
Define Russell's definition description binder, which can be read as
"the unique such that ," where ordinarily contains
as a free
variable. Our definition is meaningful only when there
is exactly one
such that is
true (see iotaval 5158);
otherwise, it evaluates to the empty set (see iotanul 5162). Russell used
the inverted iota symbol to represent the binder.
Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 5170 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | dfiota2 5148* | Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | nfiota1 5149 | Bound-variable hypothesis builder for the class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | nfiotadw 5150* | Bound-variable hypothesis builder for the class. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | nfiotaw 5151* | Bound-variable hypothesis builder for the class. (Contributed by NM, 23-Aug-2011.) |
Theorem | cbviota 5152 | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | cbviotav 5153* | Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | sb8iota 5154 | Variable substitution in description binder. Compare sb8eu 2026. (Contributed by NM, 18-Mar-2013.) |
Theorem | iotaeq 5155 | Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | iotabi 5156 | Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.) |
Theorem | uniabio 5157* | Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotaval 5158* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | iotauni 5159 | Equivalence between two different forms of . (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iotaint 5160 | Equivalence between two different forms of . (Contributed by Mario Carneiro, 24-Dec-2016.) |
Theorem | iota1 5161 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | iotanul 5162 | Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.) |
Theorem | euiotaex 5163 | Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.) |
Theorem | iotass 5164* | Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.) |
Theorem | iota4 5165 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 5166 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 5167* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 5168* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 5169 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 5170 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 5147). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 5171 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 5172* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 5173* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 5174 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 5175* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 5176 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 5177 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 5178 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 5179 | Extend the definition of a wff to include one-to-one functions. (Read: maps one-to-one into .) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27. |
Syntax | wfo 5180 | Extend the definition of a wff to include onto functions. (Read: maps onto .) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27. |
Syntax | wf1o 5181 | Extend the definition of a wff to include one-to-one onto functions. (Read: maps one-to-one onto .) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27. |
Syntax | cfv 5182 | Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .") |
Syntax | wiso 5183 | Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .) |
Definition | df-fun 5184 | Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 5214). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4037 with the maps-to notation (see df-mpt 4039). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5185), a function with a given domain and codomain (df-f 5186), a one-to-one function (df-f1 5187), an onto function (df-fo 5188), or a one-to-one onto function (df-f1o 5189). For alternate definitions, see dffun2 5192, dffun4 5193, dffun6 5196, dffun7 5209, dffun8 5210, and dffun9 5211. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fn 5185 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f 5186 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f1 5187 | Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.) |
Definition | df-fo 5188 | Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.) |
Definition | df-f1o 5189 | Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.) |
Definition | df-fv 5190* | Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 4039), but this is not required. For example, . We will later define two-argument functions using ordered pairs as . This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar notation for a function's value at , i.e., " of ," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use . (Revised by Scott Fenton, 6-Oct-2017.) |
Definition | df-isom 5191* | Define the isomorphism predicate. We read this as " is an , isomorphism of onto ." Normally, and are ordering relations on and respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that and are subscripts. (Contributed by NM, 4-Mar-1997.) |
Theorem | dffun2 5192* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun4 5193* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun5r 5194* | A way of proving a relation is a function, analogous to mo2r 2065. (Contributed by Jim Kingdon, 27-May-2020.) |
Theorem | dffun6f 5195* | 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.) |
Theorem | dffun6 5196* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
Theorem | funmo 5197* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
Theorem | dffun4f 5198* | Definition of function like dffun4 5193 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
Theorem | funrel 5199 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
Theorem | 0nelfun 5200 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
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