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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iotaval 5201* | Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | iotauni 5202 |
Equivalence between two different forms of ![]() |
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Theorem | iotaint 5203 |
Equivalence between two different forms of ![]() |
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Theorem | iota1 5204 | Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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Theorem | iotanul 5205 |
Theorem 8.22 in [Quine] p. 57. This theorem is
the result if there
isn't exactly one ![]() ![]() |
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Theorem | euiotaex 5206 |
Theorem 8.23 in [Quine] p. 58, with existential
uniqueness condition
added. This theorem proves the existence of the ![]() |
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Theorem | iotass 5207* | 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.) |
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Theorem | iota4 5208 | Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
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Theorem | iota4an 5209 | Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
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Theorem | iota5 5210* | A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
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Theorem | iotabidv 5211* | Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.) |
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Theorem | iotabii 5212 | Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.) |
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Theorem | iotacl 5213 |
Membership law for descriptions.
This can useful for expanding an unbounded iota-based definition (see df-iota 5190). (Contributed by Andrew Salmon, 1-Aug-2011.) |
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Theorem | iota2df 5214 |
A condition that allows us to represent "the unique element such that
![]() ![]() |
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Theorem | iota2d 5215* |
A condition that allows us to represent "the unique element such that
![]() ![]() |
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Theorem | eliota 5216* | An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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Theorem | eliotaeu 5217 | An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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Theorem | iota2 5218* |
The unique element such that ![]() |
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Theorem | sniota 5219 | A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
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Theorem | iotam 5220* |
Representation of "the unique element such that ![]() ![]() ![]() |
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Theorem | csbiotag 5221* | Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.) |
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Syntax | wfun 5222 |
Extend the definition of a wff to include the function predicate. (Read:
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Syntax | wfn 5223 |
Extend the definition of a wff to include the function predicate with a
domain. (Read: ![]() ![]() |
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Syntax | wf 5224 |
Extend the definition of a wff to include the function predicate with
domain and codomain. (Read: ![]() ![]() ![]() |
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Syntax | wf1 5225 |
Extend the definition of a wff to include one-to-one functions. (Read:
![]() ![]() ![]() |
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Syntax | wfo 5226 |
Extend the definition of a wff to include onto functions. (Read: ![]() ![]() ![]() |
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Syntax | wf1o 5227 |
Extend the definition of a wff to include one-to-one onto functions.
(Read: ![]() ![]() ![]() |
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Syntax | cfv 5228 |
Extend the definition of a class to include the value of a function.
(Read: The value of ![]() ![]() ![]() ![]() |
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Syntax | wiso 5229 |
Extend the definition of a wff to include the isomorphism property.
(Read: ![]() ![]() ![]() ![]() ![]() |
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Definition | df-fun 5230 |
Define predicate that determines if some class ![]() ![]() ![]() |
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Definition | df-fn 5231 | Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-f 5232 | Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-f1 5233 | 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.) |
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Definition | df-fo 5234 | 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.) |
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Definition | df-f1o 5235 | 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.) |
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Definition | df-fv 5236* |
Define the value of a function, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-isom 5237* |
Define the isomorphism predicate. We read this as "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dffun2 5238* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) |
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Theorem | dffun4 5239* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
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Theorem | dffun5r 5240* | A way of proving a relation is a function, analogous to mo2r 2088. (Contributed by Jim Kingdon, 27-May-2020.) |
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Theorem | dffun6f 5241* | 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.) |
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Theorem | dffun6 5242* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) |
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Theorem | funmo 5243* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) |
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Theorem | dffun4f 5244* | Definition of function like dffun4 5239 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.) |
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Theorem | funrel 5245 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
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Theorem | 0nelfun 5246 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
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Theorem | funss 5247 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
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Theorem | funeq 5248 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funeqi 5249 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | funeqd 5250 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
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Theorem | nffun 5251 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
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Theorem | sbcfung 5252 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
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Theorem | funeu 5253* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | funeu2 5254* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
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Theorem | dffun7 5255* | 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 5256 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
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Theorem | dffun8 5256* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 5255. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
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Theorem | dffun9 5257* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
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Theorem | funfn 5258 | An equivalence for the function predicate. (Contributed by NM, 13-Aug-2004.) |
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Theorem | funfnd 5259 | A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
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Theorem | funi 5260 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. (Contributed by NM, 30-Apr-1998.) |
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Theorem | nfunv 5261 | The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
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Theorem | funopg 5262 | 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.) |
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Theorem | funopab 5263* | 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.) |
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Theorem | funopabeq 5264* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
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Theorem | funopab4 5265* | A class of ordered pairs of values in the form used by df-mpt 4078 is a function. (Contributed by NM, 17-Feb-2013.) |
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Theorem | funmpt 5266 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
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Theorem | funmpt2 5267 | Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
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Theorem | funco 5268 | 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.) |
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Theorem | funres 5269 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funssres 5270 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
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Theorem | fun2ssres 5271 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
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Theorem | funun 5272 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
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Theorem | funcnvsn 5273 |
The converse singleton of an ordered pair is a function. This is
equivalent to funsn 5276 via cnvsn 5123, but stating it this way allows us to
skip the sethood assumptions on ![]() ![]() |
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Theorem | funsng 5274 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
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Theorem | fnsng 5275 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
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Theorem | funsn 5276 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
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Theorem | funinsn 5277 |
A function based on the singleton of an ordered pair. Unlike funsng 5274,
this holds even if ![]() ![]() |
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Theorem | funprg 5278 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) |
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Theorem | funtpg 5279 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
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Theorem | funpr 5280 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
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Theorem | funtp 5281 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
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Theorem | fnsn 5282 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | fnprg 5283 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | fntpg 5284 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
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Theorem | fntp 5285 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
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Theorem | fun0 5286 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
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Theorem | funcnvcnv 5287 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
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Theorem | funcnv2 5288* | A simpler equivalence for single-rooted (see funcnv 5289). (Contributed by NM, 9-Aug-2004.) |
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Theorem | funcnv 5289* |
The converse of a class is a function iff the class is single-rooted,
which means that for any ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | funcnv3 5290* | A condition showing a class is single-rooted. (See funcnv 5289). (Contributed by NM, 26-May-2006.) |
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Theorem | funcnveq 5291* | Another way of expressing that a class is single-rooted. Counterpart to dffun2 5238. (Contributed by Jim Kingdon, 24-Dec-2018.) |
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Theorem | fun2cnv 5292* |
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 ![]() |
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Theorem | svrelfun 5293 | A single-valued relation is a function. (See fun2cnv 5292 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
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Theorem | fncnv 5294* | Single-rootedness (see funcnv 5289) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.) |
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Theorem | fun11 5295* |
Two ways of stating that ![]() |
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Theorem | fununi 5296* | The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.) |
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Theorem | funcnvuni 5297* | The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5289 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.) |
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Theorem | fun11uni 5298* | 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.) |
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Theorem | funin 5299 | 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.) |
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Theorem | funres11 5300 | The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) |
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