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Theorem List for New Foundations Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnima 5201 The image of a function's domain is its range. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 4-Nov-2004.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfn0 5202 A function with empty domain is empty. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 15-Apr-1998.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfnimadisj 5203 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Theoremiunfopab 5204* Two ways to express a function as a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Sep-2011.) (Contributed by set.mm contributors, 19-Dec-2008.)

Theoremfnopabg 5205* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremfnopab2g 5206* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Mar-2006.)

Theoremfnopab 5207* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 5-Mar-1996.)

Theoremfnopab2 5208* Functionality and domain of an ordered-pair class abstraction. (Contributed by set.mm contributors, 29-Jan-2004.)

Theoremdmopab2 5209* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by set.mm contributors, 6-Sep-2005.)

Theoremfeq1 5210 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremfeq2 5211 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremfeq3 5212 Equality theorem for functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremfeq23 5213 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeq1d 5214 Equality deduction for functions. (Contributed by set.mm contributors, 19-Feb-2008.)

Theoremfeq2d 5215 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq12d 5216 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq1i 5217 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq2i 5218 Equality inference for functions. (Contributed by set.mm contributors, 5-Sep-2011.)

Theoremfeq23i 5219 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq23d 5220 Equality deduction for functions. (Contributed by set.mm contributors, 8-Jun-2013.)

Theoremnff 5221 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremelimf 5222 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3703, when a special case is provable, in order to convert from a hypothesis to an antecedent. (Contributed by set.mm contributors, 24-Aug-2006.)

Theoremffn 5223 A mapping is a function. (Contributed by set.mm contributors, 2-Aug-1994.)

Theoremdffn2 5224 Any function is a mapping into . (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 31-Oct-1995.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremffun 5225 A mapping is a function. (Contributed by set.mm contributors, 3-Aug-1994.)

Theoremfdm 5226 The domain of a mapping. (Contributed by set.mm contributors, 2-Aug-1994.)

Theoremfdmi 5227 The domain of a mapping. (Contributed by set.mm contributors, 28-Jul-2008.)

Theoremfrn 5228 The range of a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)

Theoremdffn3 5229 A function maps to its range. (Contributed by set.mm contributors, 1-Sep-1999.)

Theoremfss 5230 Expanding the codomain of a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfco 5231 Composition of two mappings. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 29-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfssxp 5232 A mapping is a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfunssxp 5233 Two ways of specifying a partial function from to . (Contributed by set.mm contributors, 13-Nov-2007.)

Theoremffdm 5234 A mapping is a partial function. (Contributed by set.mm contributors, 25-Nov-2007.)

Theoremopelf 5235 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by set.mm contributors, 9-Jan-2015.)

Theoremfun 5236 The union of two functions with disjoint domains. (Contributed by set.mm contributors, 22-Sep-2004.)

Theoremfnfco 5237 Composition of two functions. (Contributed by set.mm contributors, 22-May-2006.)

Theoremfssres 5238 Restriction of a function with a subclass of its domain. (Contributed by set.mm contributors, 23-Sep-2004.)

Theoremfssres2 5239 Restriction of a restricted function with a subclass of its domain. (Contributed by set.mm contributors, 21-Jul-2005.)

Theoremfcoi1 5240 Composition of a mapping and restricted identity. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfcoi2 5241 Composition of restricted identity and a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 13-Dec-2003.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfeu 5242* There is exactly one value of a function in its codomain. (Contributed by set.mm contributors, 10-Dec-2003.)

Theoremfcnvres 5243 The converse of a restriction of a function. (Contributed by set.mm contributors, 26-Mar-1998.)

Theoremfimacnvdisj 5244 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfint 5245* Function into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Oct-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfin 5246 Mapping into an intersection. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Sep-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremdmfex 5247 If a mapping is a set, its domain is a set. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 27-Aug-2006.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremf0 5248 The empty function. (Contributed by set.mm contributors, 14-Aug-1999.)

Theoremf00 5249 A class is a function with empty codomain iff it and its domain are empty. (Contributed by set.mm contributors, 10-Dec-2003.)

Theoremfconst 5250 A cross product with a singleton is a constant function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 14-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.)

Theoremfconstg 5251 A cross product with a singleton is a constant function. (Contributed by set.mm contributors, 19-Oct-2004.)

Theoremfnconstg 5252 A cross product with a singleton is a constant function. (Contributed by set.mm contributors, 24-Jul-2014.)

Theoremf1eq1 5253 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremf1eq2 5254 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremf1eq3 5255 Equality theorem for one-to-one functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremnff1 5256 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5257* Alternate definition of a one-to-one function. (Contributed by set.mm contributors, 31-Dec-1996.) (Revised by set.mm contributors, 22-Sep-2004.)

Theoremf1f 5258 A one-to-one mapping is a mapping. (Contributed by set.mm contributors, 31-Dec-1996.)

Theoremf1fn 5259 A one-to-one mapping is a function on its domain. (Contributed by set.mm contributors, 8-Mar-2014.)

Theoremf1fun 5260 A one-to-one mapping is a function. (Contributed by set.mm contributors, 8-Mar-2014.)

Theoremf1dm 5261 The domain of a one-to-one mapping. (Contributed by set.mm contributors, 8-Mar-2014.)

Theoremf1ss 5262 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1funfun 5263 Two ways to express that a set is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by set.mm contributors, 13-Aug-2004.) (Revised by Scott Fenton, 18-Apr-2021.)

Theoremf1co 5264 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 28-May-1998.)

Theoremfoeq1 5265 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremfoeq2 5266 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremfoeq3 5267 Equality theorem for onto functions. (Contributed by set.mm contributors, 1-Aug-1994.)

Theoremnffo 5268 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5269 An onto mapping is a mapping. (Contributed by set.mm contributors, 3-Aug-1994.)

Theoremfofun 5270 An onto mapping is a function. (Contributed by set.mm contributors, 29-Mar-2008.)

Theoremfofn 5271 An onto mapping is a function on its domain. (Contributed by set.mm contributors, 16-Dec-2008.)

Theoremforn 5272 The codomain of an onto function is its range. (Contributed by set.mm contributors, 3-Aug-1994.)

Theoremdffo2 5273 Alternate definition of an onto function. (Contributed by set.mm contributors, 22-Mar-2006.)

Theoremfoima 5274 The image of the domain of an onto function. (Contributed by set.mm contributors, 29-Nov-2002.)

Theoremdffn4 5275 A function maps onto its range. (Contributed by set.mm contributors, 10-May-1998.)

Theoremfunforn 5276 A function maps its domain onto its range. (Contributed by set.mm contributors, 23-Jul-2004.)

Theoremfodmrnu 5277 An onto function has unique domain and range. (Contributed by set.mm contributors, 5-Nov-2006.)

Theoremfores 5278 Restriction of a function. (Contributed by set.mm contributors, 4-Mar-1997.)

Theoremfoco 5279 Composition of onto functions. (Contributed by set.mm contributors, 22-Mar-2006.)

Theoremfoconst 5280 A nonzero constant function is onto. (Contributed by set.mm contributors, 12-Jan-2007.)

Theoremf1oeq1 5281 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremf1oeq2 5282 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremf1oeq3 5283 Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Theoremf1oeq23 5284 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremnff1o 5285 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5286 A one-to-one onto mapping is a one-to-one mapping. (Contributed by set.mm contributors, 12-Dec-2003.)

Theoremf1of 5287 A one-to-one onto mapping is a mapping. (Contributed by set.mm contributors, 12-Dec-2003.)

Theoremf1ofn 5288 A one-to-one onto mapping is function on its domain. (Contributed by set.mm contributors, 12-Dec-2003.)

Theoremf1ofun 5289 A one-to-one onto mapping is a function. (Contributed by set.mm contributors, 12-Dec-2003.)

Theoremf1odm 5290 The domain of a one-to-one onto mapping. (Contributed by set.mm contributors, 8-Mar-2014.)

Theoremdff1o2 5291 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremdff1o3 5292 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremf1ofo 5293 A one-to-one onto function is an onto function. (Contributed by set.mm contributors, 28-Apr-2004.)

Theoremdff1o4 5294 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 25-Mar-1998.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremdff1o5 5295 Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremf1orn 5296 A one-to-one function maps onto its range. (Contributed by set.mm contributors, 13-Aug-2004.)

Theoremf1f1orn 5297 A one-to-one function maps one-to-one onto its range. (Contributed by set.mm contributors, 4-Sep-2004.)

Theoremf1ocnvb 5298 A class is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by set.mm contributors, 8-Dec-2003.) (Modified by Scott Fenton, 17-Apr-2021.)

Theoremf1ocnv 5299 The converse of a one-to-one onto function is also one-to-one onto. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 11-Feb-1997.) (Revised by set.mm contributors, 22-Oct-2011.)

Theoremf1ores 5300 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by set.mm contributors, 25-Mar-1998.)

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