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Theorem List for Metamath Proof Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfunfvbrb 6001 Two ways to say that is in the domain of . (Contributed by Mario Carneiro, 1-May-2014.)

Theoremfvimacnvi 6002 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)

Theoremfvimacnv 6003 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 5668 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass3 6004 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 6003 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Theoremfunimass5 6005* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)

Theoremfunconstss 6006* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)

TheoremfvimacnvALT 6007 Alternate proof of fvimacnv 6003, based on funimass3 6004. If funimass3 6004 is ever proved directly, as opposed to using funimacnv 5666 pointwise, then the proof of funimacnv 5666 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremelpreima 6008 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfniniseg 6009 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 6010* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfniniseg2 6011* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnniniseg2OLD 6012* Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) Obsolete version of suppvalfn 6924 as of 22-Apr-2019. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremrexsuppOLD 6013* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) Obsolete version of rexsupp 6936 as of 27-May-2019. ( (New usage is discouraged.) (Proof modification is discouraged.)

Theoremunpreima 6014 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoreminpreima 6015 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)

Theoremdifpreima 6016 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)

Theoremrespreima 6017 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiinpreima 6018* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)

Theoremintpreima 6019* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)

Theoremfimacnv 6020 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

TheoremsuppssOLD 6021* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppss 6948 as of 28-May-2019. (New usage is discouraged.) (Proof modification is discouraged.)

TheoremsuppssrOLD 6022 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) Obsolete version of suppssr 6949 as of 28-May-2019. (New usage is discouraged.)

Theoremfvn0ssdmfun 6023* If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 5904. (Contributed by AV, 27-Jan-2020.)

Theoremfnopfv 6024 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)

Theoremfvelrn 6025 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)

Theoremnelrnfvne 6026 A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)

Theoremfveqdmss 6027* If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)

Theoremfveqressseq 6028* If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)

Theoremfnfvelrn 6029 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Theoremffvelrn 6030 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)

Theoremffvelrni 6031 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)

Theoremffvelrnda 6032 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremffvelrnd 6033 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremrexrn 6034* Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremralrn 6035* Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.)

Theoremelrnrexdm 6036* 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 6037* 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 6038* 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.)

Theoremeldmrexrnb 6039* For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5602 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5602 of the value of a function, may mean that the value of at is the empty set or that is not defined at . (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Theoremfvcofneq 6040* The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.)

Theoremralrnmpt 6041* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexrnmpt 6042* A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremf0cli 6043 Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.)

Theoremdff2 6044 Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)

Theoremdff3 6045* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdff4 6046* Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo3 6047* An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.)

Theoremdffo4 6048* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremdffo5 6049* Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)

Theoremexfo 6050* A relation equivalent to the existence of an onto mapping. The right-hand is not necessarily a function. (Contributed by NM, 20-Mar-2007.)

Theoremfoelrn 6051* Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)

Theoremfoco2 6052 If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)

Theoremfmpt 6053* Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremf1ompt 6054* Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)

Theoremfmpti 6055* Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremfmptd 6056* Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.)

Theoremfmptdf 6057* A version of fmptd 6056 using bound-variable hypothesis instead of a distinct variable condition for . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremffnfv 6058* A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)

Theoremffnfvf 6059 A function maps to a class to which all values belong. This version of ffnfv 6058 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Theoremfnfvrnss 6060* An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.)

Theoremrnmptss 6061* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)

Theoremfmpt2d 6062* Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.)

Theoremffvresb 6063* A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)

Theoremf1oresrab 6064* Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)

Theoremfmptco 6065* Composition of two functions expressed as ordered-pair class abstractions. If has the equation and the equation then has the equation . (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.)

Theoremfmptcof 6066* Version of fmptco 6065 where needn't be distinct from . (Contributed by NM, 27-Dec-2014.)

Theoremfmptcos 6067* Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfcompt 6068* Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremfcoconst 6069 Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Theoremfsn 6070 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)

Theoremfsng 6071 A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)

Theoremfsn2 6072 A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.)

Theoremxpsng 6073 The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.)

Theoremxpsn 6074 The Cartesian product of two singletons. (Contributed by NM, 4-Nov-2006.)

Theoremf1o2sn 6075 A singleton with a nested ordered pair is a 1-1 function of the cartesian product of two singleton onto a singleton. (Contributed by AV, 15-Aug-2019.)

Theoremresidpr 6076 Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.)

Theoremdfmpt 6077 Alternate definition for the "maps to" notation df-mpt 4517 (although it requires that be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)

Theoremfnasrn 6078 A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremressnop0 6079 If is not in , then the restriction of a singleton of to is null. (Contributed by Scott Fenton, 15-Apr-2011.)

Theoremfpr 6080 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremfprg 6081 A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)

Theoremftpg 6082 A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theoremftp 6083 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by Alexander van der Vekens, 23-Jan-2018.)

Theoremfnressn 6084 A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfunressn 6085 A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremfressnfv 6086 The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)

Theoremfvrnressn 6087 If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Theoremfvressn 6088 The value of a function restricted to the singleton containing the argument equals the value of the function for this argument. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Theoremfvn0fvelrn 6089 If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Theoremfvconst 6090 The value of a constant function. (Contributed by NM, 30-May-1999.)

Theoremfnsnb 6091 A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)

Theoremfmptsn 6092* Express a singleton function in maps-to notation. (Contributed by NM, 6-Jun-2006.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 28-Feb-2015.)

Theoremfmptsng 6093* Express a singleton function in maps-to notation. Version of fmptsn 6092 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)

Theoremfmptsnd 6094* Express a singleton function in maps-to notation. Deduction form of fmptsng 6093. (Contributed by AV, 4-Aug-2019.)

Theoremfmptap 6095* Append an additional value to a function. (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremfmptapd 6096* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfmptpr 6097* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Theoremfvresi 6098 The value of a restricted identity function. (Contributed by NM, 19-May-2004.)

Theoremfninfp 6099* Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremfnelfp 6100 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)

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