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

Theoremf1co 5101 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)

Theoremfoeq1 5102 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq2 5103 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremfoeq3 5104 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Theoremnffo 5105 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)

Theoremfof 5106 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)

Theoremfofun 5107 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)

Theoremfofn 5108 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)

Theoremforn 5109 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)

Theoremdffo2 5110 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)

Theoremfoima 5111 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Theoremdffn4 5112 A function maps onto its range. (Contributed by NM, 10-May-1998.)

Theoremfunforn 5113 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)

Theoremfodmrnu 5114 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)

Theoremfores 5115 Restriction of a function. (Contributed by NM, 4-Mar-1997.)

Theoremfoco 5116 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Theoremf1oeq1 5117 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq2 5118 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq3 5119 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Theoremf1oeq23 5120 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Theoremf1eq123d 5121 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremfoeq123d 5122 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremf1oeq123d 5123 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)

Theoremnff1o 5124 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)

Theoremf1of1 5125 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1of 5126 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofn 5127 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)

Theoremf1ofun 5128 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)

Theoremf1orel 5129 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)

Theoremf1odm 5130 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)

Theoremdff1o2 5131 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o3 5132 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1ofo 5133 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)

Theoremdff1o4 5134 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremdff1o5 5135 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremf1orn 5136 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Theoremf1f1orn 5137 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)

Theoremf1oabexg 5138* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Theoremf1ocnv 5139 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 5140 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 5141 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Theoremf1orescnv 5142 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Theoremf1imacnv 5143 Preimage of an image. (Contributed by NM, 30-Sep-2004.)

Theoremfoimacnv 5144 A reverse version of f1imacnv 5143. (Contributed by Jeff Hankins, 16-Jul-2009.)

Theoremfoun 5145 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremf1oun 5146 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)

Theoremfun11iun 5147* 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 5148 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 5149 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)

Theoremf1cnv 5150 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)

Theoremfuncocnv2 5151 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfococnv2 5152 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)

Theoremf1ococnv2 5153 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 5154 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremf1ococnv1 5155 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 5156 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Theoremfuncoeqres 5157 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremffoss 5158* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)

Theoremf11o 5159* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)

Theoremf10 5160 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)

Theoremf1o00 5161 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)

Theoremfo00 5162 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremf1o0 5163 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)

Theoremf1oi 5164 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 5165 The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)

Theoremf1osn 5166 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 5167 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1oprg 5168 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 5169* 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 5170* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Theorembrprcneu 5171* 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 5172 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)

Theoremfv2 5173* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdffv3g 5174* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremdffv4g 5175* The previous definition of function value, from before the operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 4694), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Theoremelfv 5176* Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Theoremfveq1 5177 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq2 5178 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq1i 5179 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq1d 5180 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq2i 5181 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)

Theoremfveq2d 5182 Equality deduction for function value. (Contributed by NM, 29-May-1999.)

Theoremfveq12i 5183 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)

Theoremfveq12d 5184 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremnffv 5185 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvmpt1 5186* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremnffvd 5187 Deduction version of bound-variable hypothesis builder nffv 5185. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfunfveu 5188* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremfvss 5189* The value of a function is a subset of if every element that could be a candidate for the value is a subset of . (Contributed by Mario Carneiro, 24-May-2019.)

Theoremfvssunirng 5190 The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremrelfvssunirn 5191 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)

Theoremfunfvex 5192 The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremrelrnfvex 5193 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)

Theoremfvexg 5194 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremfvex 5195 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremsefvex 5196 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
Se

Theoremfv3 5197* Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvres 5198 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Theoremfunssfv 5199 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Theoremtz6.12-1 5200* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

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