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Theorem List for Intuitionistic Logic Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1osng 5401 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Theoremf1sng 5402 A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)

Theoremfsnd 5403 A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)

Theoremf1oprg 5404 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 5405* 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 5406* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)

Theorembrprcneu 5407* If is a proper class and is any class, then there is no unique set which is related to through the binary relation . (Contributed by Scott Fenton, 7-Oct-2017.)

Theoremfvprc 5408 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)

Theoremfv2 5409* 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 5410* A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremdffv4g 5411* 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 4903), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)

Theoremelfv 5412* Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Theoremfveq1 5413 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq2 5414 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)

Theoremfveq1i 5415 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq1d 5416 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)

Theoremfveq2i 5417 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)

Theoremfveq2d 5418 Equality deduction for function value. (Contributed by NM, 29-May-1999.)

Theorem2fveq3 5419 Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)

Theoremfveq12i 5420 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)

Theoremfveq12d 5421 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)

Theoremfveqeq2d 5422 Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)

Theoremfveqeq2 5423 Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)

Theoremnffv 5424 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnffvmpt1 5425* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)

Theoremnffvd 5426 Deduction version of bound-variable hypothesis builder nffv 5424. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfunfveu 5427* A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)

Theoremfvss 5428* 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 5429 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 5430 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 5431 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 5432 If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)

Theoremfvexg 5433 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremfvex 5434 Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)

Theoremsefvex 5435 If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
Se

Theoremfvifdc 5436 Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
DECID

Theoremfv3 5437* 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 5438 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)

Theoremfvresd 5439 The value of a restricted function, deduction version of fvres 5438. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Theoremfunssfv 5440 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)

Theoremtz6.12-1 5441* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremtz6.12 5442* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)

Theoremtz6.12f 5443* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)

Theoremtz6.12c 5444* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Theoremndmfvg 5445 The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)

Theoremrelelfvdm 5446 If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)

Theoremnfvres 5447 The value of a non-member of a restriction is the empty set. (Contributed by NM, 13-Nov-1995.)

Theoremnfunsn 5448 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 5449 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)

Theoremcsbfv12g 5450 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)

Theoremcsbfv2g 5451* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Theoremcsbfvg 5452* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)

Theoremfunbrfv 5453 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 5454 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)

Theoremfnbrfvb 5455 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5456 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)

Theoremfunbrfvb 5457 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Theoremfunopfvb 5458 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Theoremfunbrfv2b 5459 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5im 5460* 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 5156 and dmmptss 5030. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfnrnfv 5461* 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 5462* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Theoremdfimafn 5463* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5464* 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 5465* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5466* 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 5467* Deduction form of dffn5im 5460. (Contributed by Mario Carneiro, 8-Jan-2015.)

Theoremfeqresmpt 5468* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdffn5imf 5469* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfvelimab 5470* 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 5471 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfniinfv 5472* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Theoremfnsnfv 5473 Singleton of function value. (Contributed by NM, 22-May-1998.)

Theoremfnimapr 5474 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremssimaex 5475* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Theoremssimaexg 5476* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfvdm 5477 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2 5478* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2f 5479 The value of a function. Version of funfvdm2 5478 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfvun1 5480 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5481 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdmfco 5482 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5483 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 5484 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 5485 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvco4 5486 Value of a composition. (Contributed by BJ, 7-Jul-2022.)

Theoremfvopab3g 5487* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5488* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptss2 5489* 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 5490* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt 5491* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5492* 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 5493* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5494* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5495* Deduction version of fvmpt 5491. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptrcl 5496* Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)

Theoremfvmpt2 5497* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptssdm 5498* 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 5499* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfvmpt2d 5500* Deduction version of fvmpt2 5497. (Contributed by Thierry Arnoux, 8-Dec-2016.)

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