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Type | Label | Description |
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Statement | ||
Theorem | iota4 5101 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota4an 5102 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Theorem | iota5 5103* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Theorem | iotabidv 5104* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
Theorem | iotabii 5105 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | iotacl 5106 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 5083). (Contributed by Andrew Salmon, 1-Aug-2011.) |
Theorem | iota2df 5107 | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2d 5108* | A condition that allows us to represent "the unique element such that " with a class expression . (Contributed by NM, 30-Dec-2014.) |
Theorem | iota2 5109* | The unique element such that . (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Theorem | sniota 5110 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Theorem | csbiotag 5111* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
Syntax | wfun 5112 | Extend the definition of a wff to include the function predicate. (Read: is a function.) |
Syntax | wfn 5113 | Extend the definition of a wff to include the function predicate with a domain. (Read: is a function on .) |
Syntax | wf 5114 | Extend the definition of a wff to include the function predicate with domain and codomain. (Read: maps into .) |
Syntax | wf1 5115 | 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 5116 | 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 5117 | 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 5118 | Extend the definition of a class to include the value of a function. (Read: The value of at , or " of .") |
Syntax | wiso 5119 | Extend the definition of a wff to include the isomorphism property. (Read: is an , isomorphism of onto .) |
Definition | df-fun 5120 | Define predicate that determines if some class is a function. Definition 10.1 of [Quine] p. 65. For example, the expression is true (funi 5150). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 3984 with the maps-to notation (see df-mpt 3986). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5121), a function with a given domain and codomain (df-f 5122), a one-to-one function (df-f1 5123), an onto function (df-fo 5124), or a one-to-one onto function (df-f1o 5125). For alternate definitions, see dffun2 5128, dffun4 5129, dffun6 5132, dffun7 5145, dffun8 5146, and dffun9 5147. (Contributed by NM, 1-Aug-1994.) |
Definition | df-fn 5121 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
Definition | df-f 5122 | 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 5123 | 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 5124 | 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 5125 | 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 5126* | Define the value of a function, , also known as function application. For example, . Typically, function is defined using maps-to notation (see df-mpt 3986), but this is not required. For example, F = { 2 , 6 , 3 , 9 } -> ( F 3 ) = 9 . 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 5127* | 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 5128* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun4 5129* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
Theorem | dffun5r 5130* | A way of proving a relation is a function, analogous to mo2r 2049. (Contributed by Jim Kingdon, 27-May-2020.) |
Theorem | dffun6f 5131* | 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 5132* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
Theorem | funmo 5133* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
Theorem | dffun4f 5134* | Definition of function like dffun4 5129 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
Theorem | funrel 5135 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
Theorem | 0nelfun 5136 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
Theorem | funss 5137 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
Theorem | funeq 5138 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
Theorem | funeqi 5139 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | funeqd 5140 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
Theorem | nffun 5141 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
Theorem | sbcfung 5142 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Theorem | funeu 5143* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | funeu2 5144* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
Theorem | dffun7 5145* | 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 5146 shows that it doesn't matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
Theorem | dffun8 5146* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5145. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Theorem | dffun9 5147* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
Theorem | funfn 5148 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
Theorem | funfnd 5149 | A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Theorem | funi 5150 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
Theorem | nfunv 5151 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
Theorem | funopg 5152 | 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.) |
Theorem | funopab 5153* | 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.) |
Theorem | funopabeq 5154* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
Theorem | funopab4 5155* | A class of ordered pairs of values in the form used by df-mpt 3986 is a function. (Contributed by NM, 17-Feb-2013.) |
Theorem | funmpt 5156 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
Theorem | funmpt2 5157 | Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
Theorem | funco 5158 | 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.) |
Theorem | funres 5159 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
Theorem | funssres 5160 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
Theorem | fun2ssres 5161 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
Theorem | funun 5162 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
Theorem | funcnvsn 5163 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 5166 via cnvsn 5016, but stating it this way allows us to skip the sethood assumptions on and . (Contributed by NM, 30-Apr-2015.) |
Theorem | funsng 5164 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
Theorem | fnsng 5165 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Theorem | funsn 5166 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
Theorem | funinsn 5167 | A function based on the singleton of an ordered pair. Unlike funsng 5164, this holds even if or is a proper class. (Contributed by Jim Kingdon, 17-Apr-2022.) |
Theorem | funprg 5168 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
Theorem | funtpg 5169 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Theorem | funpr 5170 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
Theorem | funtp 5171 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
Theorem | fnsn 5172 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | fnprg 5173 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fntpg 5174 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
Theorem | fntp 5175 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | fun0 5176 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
Theorem | funcnvcnv 5177 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
Theorem | funcnv2 5178* | A simpler equivalence for single-rooted (see funcnv 5179). (Contributed by NM, 9-Aug-2004.) |
Theorem | funcnv 5179* | 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 5178 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
Theorem | funcnv3 5180* | A condition showing a class is single-rooted. (See funcnv 5179). (Contributed by NM, 26-May-2006.) |
Theorem | funcnveq 5181* | Another way of expressing that a class is single-rooted. Counterpart to dffun2 5128. (Contributed by Jim Kingdon, 24-Dec-2018.) |
Theorem | fun2cnv 5182* | 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.) |
Theorem | svrelfun 5183 | A single-valued relation is a function. (See fun2cnv 5182 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
Theorem | fncnv 5184* | Single-rootedness (see funcnv 5179) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.) |
Theorem | fun11 5185* | 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.) |
Theorem | fununi 5186* | The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.) |
Theorem | funcnvuni 5187* | The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5179 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.) |
Theorem | fun11uni 5188* | 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.) |
Theorem | funin 5189 | 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.) |
Theorem | funres11 5190 | The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) |
Theorem | funcnvres 5191 | The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) |
Theorem | cnvresid 5192 | Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.) |
Theorem | funcnvres2 5193 | 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.) |
Theorem | funimacnv 5194 | The image of the preimage of a function. (Contributed by NM, 25-May-2004.) |
Theorem | funimass1 5195 | A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.) |
Theorem | funimass2 5196 | A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.) |
Theorem | imadiflem 5197 | One direction of imadif 5198. This direction does not require . (Contributed by Jim Kingdon, 25-Dec-2018.) |
Theorem | imadif 5198 | The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.) |
Theorem | imainlem 5199 | One direction of imain 5200. This direction does not require . (Contributed by Jim Kingdon, 25-Dec-2018.) |
Theorem | imain 5200 | The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.) |
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