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

Theoremfunfni 5201 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)

Theoremfndmu 5202 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)

Theoremfnbr 5203 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Theoremfnop 5204 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfneu 5205* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfneu2 5206* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)

Theoremfnun 5207 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfnunsn 5208 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Theoremfnco 5209 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfnresdm 5210 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Theoremfnresdisj 5211 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)

Theorem2elresin 5212 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)

Theoremfnssresb 5213 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)

Theoremfnssres 5214 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)

Theoremfnresin1 5215 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnresin2 5216 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)

Theoremfnres 5217* An equivalence for functionality of a restriction. Compare dffun8 5139. (Contributed by Mario Carneiro, 20-May-2015.)

Theoremfnresi 5218 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)

Theoremfnima 5219 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfn0 5220 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfnimadisj 5221 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Theoremfnimaeq0 5222 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 26314. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Theoremdfmpt3 5223 Alternate definition for the "maps to" notation df-mpt 3976. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremfnopabg 5224* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Theoremfnopab 5225* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)

Theoremmptfng 5226* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)

Theoremfnmpt 5227* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)

Theoremmpt0 5228 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfnmpti 5229* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdmmpti 5230* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptun 5231 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremfeq1 5232 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq2 5233 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq3 5234 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)

Theoremfeq23 5235 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeq1d 5236 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)

Theoremfeq2d 5237 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq12d 5238 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq123d 5239 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq1i 5240 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq2i 5241 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)

Theoremfeq23i 5242 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremfeq23d 5243 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)

Theoremnff 5244 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremelimf 5245 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3511, when a special case is provable, in order to convert from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)

Theoremffn 5246 A mapping is a function. (Contributed by NM, 2-Aug-1994.)

Theoremdffn2 5247 Any function is a mapping into . (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremffun 5248 A mapping is a function. (Contributed by NM, 3-Aug-1994.)

Theoremfrel 5249 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)

Theoremfdm 5250 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)

Theoremfdmi 5251 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)

Theoremfrn 5252 The range of a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremdffn3 5253 A function maps to its range. (Contributed by NM, 1-Sep-1999.)

Theoremfss 5254 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfco 5255 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfco2 5256 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfssxp 5257 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfex2 5258 A function with bounded domain and range is a set. This version of fex 5601 is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremfunssxp 5259 Two ways of specifying a partial function from to . (Contributed by NM, 13-Nov-2007.)

Theoremffdm 5260 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)

Theoremopelf 5261 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremfun 5262 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Theoremfun2 5263 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)

Theoremfnfco 5264 Composition of two functions. (Contributed by NM, 22-May-2006.)

Theoremfssres 5265 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)

Theoremfssres2 5266 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)

Theoremfresin 5267 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremresasplit 5268 If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaun 5269 The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaunres2 5270 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfresaunres1 5271 From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)

Theoremfcoi1 5272 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfcoi2 5273 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfeu 5274* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)

Theoremfcnvres 5275 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Theoremfimacnvdisj 5276 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfint 5277* Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfin 5278 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfabexg 5279* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremfabex 5280* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Theoremdmfex 5281 If a mapping is a set, its domain is a set. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremf0 5282 The empty function. (Contributed by NM, 14-Aug-1999.)

Theoremf00 5283 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Theoremfconst 5284 A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfconstg 5285 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Theoremfnconstg 5286 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)

Theoremfconst6g 5287 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfconst6 5288 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremf1eq1 5289 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq2 5290 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq3 5291 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremnff1 5292 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5293* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Theoremf1f 5294 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Theoremf1fn 5295 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Theoremf1fun 5296 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Theoremf1rel 5297 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5298 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5299 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.)

Theoremf1ssr 5300 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

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