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Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfcnvres 5101 The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)

Theoremfimacnvdisj 5102 The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)

Theoremfintm 5103* Function into an intersection. (Contributed by Jim Kingdon, 28-Dec-2018.)

Theoremfin 5104 Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Theoremfabexg 5105* Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremfabex 5106* Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)

Theoremdmfex 5107 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 5108 The empty function. (Contributed by NM, 14-Aug-1999.)

Theoremf00 5109 A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)

Theoremfconst 5110 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 5111 A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)

Theoremfnconstg 5112 A cross product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)

Theoremfconst6g 5113 Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfconst6 5114 A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)

Theoremf1eq1 5115 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq2 5116 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1eq3 5117 Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

Theoremnff1 5118 Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)

Theoremdff12 5119* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)

Theoremf1f 5120 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)

Theoremf1fn 5121 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)

Theoremf1fun 5122 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)

Theoremf1rel 5123 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremf1dm 5124 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)

Theoremf1ss 5125 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 5126 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremf1ssres 5127 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremf1cnvcnv 5128 Two ways to express that a set (not necessarily a function) 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 NM, 13-Aug-2004.)

Theoremf1co 5129 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Theoremfoeq1 5130 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq2 5131 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq3 5132 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremnffo 5133 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5134 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremfofun 5135 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Theoremfofn 5136 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Theoremforn 5137 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)

Theoremdffo2 5138 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Theoremfoima 5139 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Theoremdffn4 5140 A function maps onto its range. (Contributed by NM, 10-May-1998.)

Theoremfunforn 5141 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Theoremfodmrnu 5142 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Theoremfores 5143 Restriction of a function. (Contributed by NM, 4-Mar-1997.)

Theoremfoco 5144 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Theoremf1oeq1 5145 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq2 5146 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5147 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5148 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5149 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5150 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5151 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremnff1o 5152 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5153 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5154 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5155 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5156 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5157 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5158 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5159 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5160 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5161 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5162 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5163 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5164 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5165 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5166* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5167 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ocnvb 5168 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)

Theoremf1ores 5169 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5170 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5171 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5172 A reverse version of f1imacnv 5171. (Contributed by Jeff Hankins, 16-Jul-2009.)

Theoremfoun 5173 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5174 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5175* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremresdif 5176 The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)

Theoremf1oco 5177 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5178 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5179 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfococnv2 5180 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1ococnv2 5181 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)

Theoremf1cocnv2 5182 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1ococnv1 5183 The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)

Theoremf1cocnv1 5184 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremfuncoeqres 5185 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremffoss 5186* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Theoremf11o 5187* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

Theoremf10 5188 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)

Theoremf1o00 5189 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Theoremfo00 5190 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremf1o0 5191 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)

Theoremf1oi 5192 A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ovi 5193 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)

Theoremf1osn 5194 A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1osng 5195 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1oprg 5196 An unordered pair of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theoremtz6.12-2 5197* Function value when is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfveu 5198* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Theorembrprcneu 5199* If is a proper class, then there is no unique binary relationship with as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)

Theoremfvprc 5200 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)

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