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

Theoremfv2 5201* 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 5202* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremdffv4g 5203* 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 4722), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Theoremelfv 5204* Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Theoremfveq1 5205 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq2 5206 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq1i 5207 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq1d 5208 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq2i 5209 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)

Theoremfveq2d 5210 Equality deduction for function value. (Contributed by NM, 29-May-1999.)

Theoremfveq12i 5211 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)

Theoremfveq12d 5212 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremnffv 5213 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvmpt1 5214* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremnffvd 5215 Deduction version of bound-variable hypothesis builder nffv 5213. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfunfveu 5216* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremfvss 5217* 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 5218 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 5219 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 5220 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 5221 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)

Theoremfvexg 5222 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremfvex 5223 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremsefvex 5224 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
Se

Theoremfv3 5225* 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 5226 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Theoremfunssfv 5227 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Theoremtz6.12-1 5228* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12 5229* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Theoremtz6.12f 5230* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Theoremtz6.12c 5231* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremndmfvg 5232 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)

Theoremrelelfvdm 5233 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)

Theoremnfvres 5234 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Theoremnfunsn 5235 If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theorem0fv 5236 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremcsbfv12g 5237 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Theoremcsbfv2g 5238* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Theoremcsbfvg 5239* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

Theoremfunbrfv 5240 The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfunopfv 5241 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)

Theoremfnbrfvb 5242 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5243 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)

Theoremfunbrfvb 5244 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Theoremfunopfvb 5245 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Theoremfunbrfv2b 5246 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5im 5247* Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4966 and dmmptss 4845. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfnrnfv 5248* The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremfvelrnb 5249* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Theoremdfimafn 5250* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5251* Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass4 5252* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5253* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfeqmptd 5254* Deduction form of dffn5im 5247. (Contributed by Mario Carneiro, 8-Jan-2015.)

Theoremfeqresmpt 5255* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdffn5imf 5256* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfvelimab 5257* Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)

Theoremfvi 5258 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfniinfv 5259* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Theoremfnsnfv 5260 Singleton of function value. (Contributed by NM, 22-May-1998.)

Theoremfnimapr 5261 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremssimaex 5262* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Theoremssimaexg 5263* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfvdm 5264 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2 5265* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2f 5266 The value of a function. Version of funfvdm2 5265 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfvun1 5267 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5268 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdmfco 5269 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5270 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)

Theoremfvco 5271 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)

Theoremfvco3 5272 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvopab3g 5273* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5274* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptss2 5275* A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)

Theoremfvmptg 5276* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt 5277* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5278* Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt3 5279* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5280* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5281* Deduction version of fvmpt 5277. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt2 5282* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptssdm 5283* If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)

Theoremmptfvex 5284* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfvmpt2d 5285* Deduction version of fvmpt2 5282. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremfvmptdf 5286* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5287* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5288* Alternate deduction version of fvmpt 5277, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5289* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5293. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5290* Closed theorem form of fvmpt 5277. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5291* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5276 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvopab6 5292* Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremeqfnfv 5293* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv2 5294* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremeqfnfv3 5295* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5296* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5297* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5293 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5298* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5299* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5300* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

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