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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fndmdif 6001* | Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | fndmdifcom 6002 | The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | fndmdifeq0 6003 | The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | fndmin 6004* | Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
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Theorem | fneqeql 6005 | Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
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Theorem | fneqeql2 6006 | Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
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Theorem | fnreseql 6007 | Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
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Theorem | chfnrn 6008* | 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.) |
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Theorem | funfvop 6009 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.) |
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Theorem | funfvbrb 6010 |
Two ways to say that ![]() ![]() |
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Theorem | fvimacnvi 6011 | A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.) |
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Theorem | fvimacnv 6012 | 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 5667 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
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Theorem | funimass3 6013 |
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 6012 would be the special case of ![]() |
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Theorem | funimass5 6014* | A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.) |
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Theorem | funconstss 6015* | Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.) |
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Theorem | fvimacnvALT 6016 | Alternate proof of fvimacnv 6012, based on funimass3 6013. If funimass3 6013 is ever proved directly, as opposed to using funimacnv 5665 pointwise, then the proof of funimacnv 5665 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | elpreima 6017 | Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | fniniseg 6018 | 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.) |
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Theorem | fncnvima2 6019* | Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | fniniseg2 6020* | Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
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Theorem | unpreima 6021 | Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | inpreima 6022 | Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.) |
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Theorem | difpreima 6023 | Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.) |
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Theorem | respreima 6024 | The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | iinpreima 6025* | Preimage of an intersection. (Contributed by FL, 16-Apr-2012.) |
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Theorem | intpreima 6026* | Preimage of an intersection. (Contributed by FL, 28-Apr-2012.) |
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Theorem | fimacnv 6027 | The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.) |
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Theorem | fvn0ssdmfun 6028* | 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 5911. (Contributed by AV, 27-Jan-2020.) |
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Theorem | fnopfv 6029 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.) |
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Theorem | fvelrn 6030 | A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
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Theorem | nelrnfvne 6031 | A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.) |
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Theorem | fveqdmss 6032* | 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.) |
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Theorem | fveqressseq 6033* | 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.) |
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Theorem | fnfvelrn 6034 | A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.) |
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Theorem | ffvelrn 6035 | A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
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Theorem | ffvelrni 6036 | A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.) |
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Theorem | ffvelrnda 6037 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
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Theorem | ffvelrnd 6038 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
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Theorem | rexrn 6039* | Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
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Theorem | ralrn 6040* | Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
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Theorem | elrnrexdm 6041* | 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.) |
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Theorem | elrnrexdmb 6042* | 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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Theorem | eldmrexrn 6043* | 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.) |
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Theorem | eldmrexrnb 6044* |
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 5597 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 5597 of the value of a function,
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Theorem | fvcofneq 6045* | The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.) |
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Theorem | ralrnmpt 6046* | A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
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Theorem | rexrnmpt 6047* | A restricted quantifier over an image set. (Contributed by Mario Carneiro, 20-Aug-2015.) |
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Theorem | f0cli 6048 | Unconditional closure of a function when the range includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.) |
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Theorem | dff2 6049 | Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.) |
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Theorem | dff3 6050* | Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.) |
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Theorem | dff4 6051* | Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.) |
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Theorem | dffo3 6052* | An onto mapping expressed in terms of function values. (Contributed by NM, 29-Oct-2006.) |
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Theorem | dffo4 6053* | Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.) |
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Theorem | dffo5 6054* | Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.) |
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Theorem | exfo 6055* |
A relation equivalent to the existence of an onto mapping. The
right-hand ![]() |
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Theorem | foelrn 6056* | Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) |
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Theorem | foco2 6057 | If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) |
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Theorem | fmpt 6058* | Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
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Theorem | f1ompt 6059* | Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.) |
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Theorem | fmpti 6060* | Functionality of the mapping operation. (Contributed by NM, 19-Mar-2005.) (Revised by Mario Carneiro, 1-Sep-2015.) |
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Theorem | fmptd 6061* | Domain and codomain of the mapping operation; deduction form. (Contributed by Mario Carneiro, 13-Jan-2013.) |
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Theorem | fmpt3d 6062* | Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.) |
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Theorem | fmptdf 6063* |
A version of fmptd 6061 using bound-variable hypothesis instead of a
distinct variable condition for ![]() |
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Theorem | ffnfv 6064* | A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.) |
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Theorem | ffnfvf 6065 | A function maps to a class to which all values belong. This version of ffnfv 6064 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.) |
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Theorem | fnfvrnss 6066* | An upper bound for range determined by function values. (Contributed by NM, 8-Oct-2004.) |
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Theorem | frnssb 6067* | A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.) |
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Theorem | rnmptss 6068* | The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
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Theorem | fmpt2d 6069* | Domain and codomain of the mapping operation; deduction form. (Contributed by NM, 27-Dec-2014.) |
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Theorem | ffvresb 6070* | A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.) |
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Theorem | f1oresrab 6071* | Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.) |
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Theorem | fmptco 6072* |
Composition of two functions expressed as ordered-pair class
abstractions. If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fmptcof 6073* |
Version of fmptco 6072 where ![]() ![]() |
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Theorem | fmptcos 6074* | Composition of two functions expressed as mapping abstractions. (Contributed by NM, 22-May-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
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Theorem | fcompt 6075* | 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.) |
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Theorem | fcoconst 6076 | Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.) |
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Theorem | fsn 6077 | A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.) |
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Theorem | fsn2 6078 | A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.) |
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Theorem | fsng 6079 | A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.) |
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Theorem | fsn2g 6080 | A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
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Theorem | xpsng 6081 | The Cartesian product of two singletons. (Contributed by Mario Carneiro, 30-Apr-2015.) |
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Theorem | xpsn 6082 | The Cartesian product of two singletons. (Contributed by NM, 4-Nov-2006.) |
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Theorem | f1o2sn 6083 | 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.) |
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Theorem | residpr 6084 | Restriction of the identity to a pair. (Contributed by AV, 11-Dec-2018.) |
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Theorem | dfmpt 6085 |
Alternate definition for the "maps to" notation df-mpt 4456 (although it
requires that ![]() |
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Theorem | fnasrn 6086 | A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
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Theorem | ressnop0 6087 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fpr 6088 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | fprg 6089 | A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.) |
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Theorem | ftpg 6090 | A function with a domain of three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
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Theorem | ftp 6091 | 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.) |
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Theorem | fnressn 6092 | A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
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Theorem | funressn 6093 | A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.) |
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Theorem | fressnfv 6094 | The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.) |
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Theorem | fvrnressn 6095 | 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.) |
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Theorem | fvressn 6096 | 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.) |
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Theorem | fvn0fvelrn 6097 | 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.) |
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Theorem | fvconst 6098 | The value of a constant function. (Contributed by NM, 30-May-1999.) |
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Theorem | fnsnb 6099 | 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.) |
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Theorem | fmptsn 6100* | 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.) |
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