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Theorem List for Intuitionistic Logic Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunopfv 5501 The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)

Theoremfnbrfvb 5502 Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfnopfvb 5503 Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)

Theoremfunbrfvb 5504 Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)

Theoremfunopfvb 5505 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)

Theoremfunbrfv2b 5506 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)

Theoremdffn5im 5507* 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 5201 and dmmptss 5075. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfnrnfv 5508* 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 5509* A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Theoremdfimafn 5510* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremdfimafn2 5511* 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 5512* Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfvelima 5513* 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 5514* Deduction form of dffn5im 5507. (Contributed by Mario Carneiro, 8-Jan-2015.)

Theoremfeqresmpt 5515* Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremdffn5imf 5516* Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)

Theoremfvelimab 5517* 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 5518 The value of the identity function. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfniinfv 5519* The indexed intersection of a function's values is the intersection of its range. (Contributed by NM, 20-Oct-2005.)

Theoremfnsnfv 5520 Singleton of function value. (Contributed by NM, 22-May-1998.)

Theoremfnimapr 5521 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)

Theoremssimaex 5522* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)

Theoremssimaexg 5523* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)

Theoremfunfvdm 5524 A simplified expression for the value of a function when we know it's a function. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2 5525* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfunfvdm2f 5526 The value of a function. Version of funfvdm2 5525 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Theoremfvun1 5527 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremfvun2 5528 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremdmfco 5529 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Theoremfvco2 5530 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 5531 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 5532 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremfvco4 5533 Value of a composition. (Contributed by BJ, 7-Jul-2022.)

Theoremfvopab3g 5534* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfvopab3ig 5535* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Theoremfvmptss2 5536* 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 5537* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfvmpt 5538* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)

Theoremfvmpts 5539* 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 5540* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremfvmpt3i 5541* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptd 5542* Deduction version of fvmpt 5538. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremmptrcl 5543* 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 5544* Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)

Theoremfvmptssdm 5545* 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 5546* Sufficient condition for a maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfvmpt2d 5547* Deduction version of fvmpt2 5544. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Theoremfvmptdf 5548* Alternate deduction version of fvmpt 5538, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv 5549* Alternate deduction version of fvmpt 5538, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremfvmptdv2 5550* Alternate deduction version of fvmpt 5538, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)

Theoremmpteqb 5551* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 5558. (Contributed by Mario Carneiro, 14-Nov-2014.)

Theoremfvmptt 5552* Closed theorem form of fvmpt 5538. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremfvmptf 5553* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 5537 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremfvmptd3 5554* Deduction version of fvmpt 5538. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Theoremelfvmptrab1 5555* 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 5556* 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 5557* 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 5558* 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 5559* 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 5560* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeqfnfvd 5561* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremeqfnfv2f 5562* 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 5558 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Theoremeqfunfv 5563* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)

Theoremfvreseq 5564* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)

Theoremfndmdif 5565* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmdifcom 5566 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfndmin 5567* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremfneqeql 5568 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremfneqeql2 5569 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Theoremfnreseql 5570 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Theoremchfnrn 5571* 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 5572 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)

Theoremfunfvbrb 5573 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 5574 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 5575 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 5241 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 5576 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 5575 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 5577* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 5578* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

Theoremelpreima 5579 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 5580 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 5581* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 5582* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2 5583* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremrexsupp 5584* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Theoremunpreima 5585 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 5586 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 5587 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 5588 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimacnv 5589 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

Theoremfnopfv 5590 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 5591 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremfnfvelrn 5592 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 5593 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 5594 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 5595 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 5596 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 5597* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 5598* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 5599* 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 5600* 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.)

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