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

Theoremssimaexg 5401* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfvdm 5402 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2 5403* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2f 5404 The value of a function. Version of funfvdm2 5403 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfvun1 5405 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5406 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdmfco 5407 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5408 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 5409 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 5410 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvco4 5411 Value of a composition. (Contributed by BJ, 7-Jul-2022.)

Theoremfvopab3g 5412* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5413* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptss2 5414* 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 5415* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt 5416* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5417* 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 5418* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5419* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5420* Deduction version of fvmpt 5416. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptrcl 5421* 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 5422* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptssdm 5423* 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 5424* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfvmpt2d 5425* Deduction version of fvmpt2 5422. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremfvmptdf 5426* Alternate deduction version of fvmpt 5416, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5427* Alternate deduction version of fvmpt 5416, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5428* Alternate deduction version of fvmpt 5416, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5429* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5436. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5430* Closed theorem form of fvmpt 5416. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5431* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5415 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptd3 5432* Deduction version of fvmpt 5416. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Theoremelfvmptrab1 5433* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Theoremelfvmptrab 5434* Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Theoremfvopab6 5435* 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 5436* 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 5437* 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 5438* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5439* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5440* 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 5436 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5441* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5442* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5443* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5444 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmin 5445* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5446 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5447 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5448 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5449* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)

Theoremfunfvop 5450 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5451 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5452 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5453 The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5126 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5454 A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5453 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5455* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5456* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

Theoremelpreima 5457 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5458 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfncnvima2 5459* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5460* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2 5461* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5462* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5463 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5464 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5465 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5466 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimacnv 5467 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremfnopfv 5468 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5469 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5470 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5471 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5472 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5473 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5474 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5475* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5476* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 5477* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremelrnrexdmb 5478* For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Theoremeldmrexrn 5479* For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Theoremralrnmpt 5480* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 5481* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremdff2 5482 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3im 5483* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdff4im 5484* Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)

Theoremdffo3 5485* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 5486* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 5487* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremfmpt 5488* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 5489* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 5490* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 5491* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfmpttd 5492* Version of fmptd 5491 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.)

Theoremfmpt3d 5493* Domain and codomain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Theoremfmptdf 5494* A version of fmptd 5491 using bound-variable hypothesis instead of a distinct variable condition for . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremffnfv 5495* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 5496 A function maps to a class to which all values belong. This version of ffnfv 5495 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 5497* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremrnmptss 5498* The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremfmpt2d 5499* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremffvresb 5500* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

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