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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremsymdifV 25701 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
(++)

Theoremsymdifid 25702 Symmetric difference yields the empty class with the same argument twice. (Contributed by Scott Fenton, 25-Apr-2012.)
(++)

Theoremsymdifass 25703 Symmetric difference associates. (Contributed by Scott Fenton, 24-Apr-2012.)
(++)(++) (++)(++)

Theorembrsymdif 25704 The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.)
(++)

19.7.38  Quantifier-free definitions

Syntaxctxp 25705 Declare the syntax for tail cross product.

Syntaxcpprod 25706 Declare the syntax for the parallel product.
pprod

Syntaxcsset 25707 Declare the subset relationship class.

Syntaxctrans 25708 Declare the transitive set class.

Syntaxcbigcup 25709 Declare the set union relationship.

Syntaxcfix 25710 Declare the syntax for the fixpoints of a class.

Syntaxclimits 25711 Declare the class of limit ordinals.

Syntaxcfuns 25712 Declare the syntax for the class of all function.

Syntaxcsingle 25713 Declare the syntax for the singleton function.
Singleton

Syntaxcsingles 25714 Declare the syntax for the class of all singletons.

Syntaxcimage 25715 Declare the syntax for the image functor.
Image

Syntaxccart 25716 Declare the syntax for the cartesian function.
Cart

Syntaxcimg 25717 Declare the syntax for the image function.
Img

Syntaxcdomain 25718 Declare the syntax for the domain function.
Domain

Syntaxcrange 25719 Declare the syntax for the range function.
Range

Syntaxcapply 25720 Declare the syntax for the application function.
Apply

Syntaxccup 25721 Declare the syntax for the cup function.
Cup

Syntaxccap 25722 Declare the syntax for the cap function.
Cap

Syntaxcsuccf 25723 Declare the syntax for the successor function.
Succ

Syntaxcfunpart 25724 Declare the syntax for the functional part functor.
Funpart

Syntaxcfullfn 25725 Declare the syntax for the full function functor.
FullFun

Syntaxcrestrict 25726 Declare the syntax for the restriction function.
Restrict

Syntaxcub 25727 Declare the syntax for the upper bound relationship functor.
UB

Syntaxclb 25728 Declare the syntax for the lower bound relationship functor.
LB

Definitiondf-txp 25729 Define the tail cross of two classes. Membership in this class is defined by txpss3v 25754 and brtxp 25756. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-pprod 25730 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 25760 and brpprod 25761. (Contributed by Scott Fenton, 11-Apr-2014.)
pprod

Definitiondf-sset 25731 Define the subset class. For the value, see brsset 25765. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-trans 25732 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-bigcup 25733 Define the Bigcup function, which, per fvbigcup 25778, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
(++)

Definitiondf-fix 25734 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Definitiondf-limits 25735 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Definitiondf-funs 25736 Define the class of all functions. See elfuns 25791 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)

Definitiondf-singleton 25737 Define the singleton function. See brsingle 25793 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton (++)

Definitiondf-singles 25738 Define the class of all singletons. See elsingles 25794 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singleton

Definitiondf-image 25739 Define the image functor. This function takes a set to a function , providing that the latter exists. See imageval 25806 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image (++)

Definitiondf-cart 25740 Define the cartesian product function. See brcart 25808 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart (++)pprod

Definitiondf-img 25741 Define the image function. See brimg 25813 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img Image Cart

Definitiondf-domain 25742 Define the domain function. See brdomain 25809 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain Image

Definitiondf-range 25743 Define the range function. See brrange 25810 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range Image

Definitiondf-cup 25744 Define the little cup function. See brcup 25815 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup (++)

Definitiondf-cap 25745 Define the little cap function. See brcap 25816 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap (++)

Definitiondf-restrict 25746 Define the restriction function. See brrestrict 25825 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict Cap Cart Range

Definitiondf-succf 25747 Define the successor function. See brsuccf 25817 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ Cup Singleton

Definitiondf-apply 25748 Define the application function. See brapply 25814 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Apply (++) Singleton Img pprod Singleton

Definitiondf-funpart 25749 Define the functional part of a class . This is the maximal part of that is a function. See funpartfun 25819 and funpartfv 25821 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart Image Singleton

Definitiondf-fullfun 25750 Define the full function over . This is a function with domain that always agrees with for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
FullFun Funpart Funpart

Definitiondf-ub 25751 Define the upper bound relationship functor. See brub 25830 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB

Definitiondf-lb 25752 Define the lower bound relationship functor. See brlb 25831 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB UB

Theorembrv 25753 The binary relationship over always holds. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremtxpss3v 25754 A tail cross product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremtxprel 25755 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp 25756 Characterize a trinary relationship over a tail cross product. Together with txpss3v 25754, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp2 25757* The binary relationship over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremdfpprod2 25758 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theorempprodcnveq 25759 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod pprod

Theorempprodss4v 25760 The parallel product is a subclass of . (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod 25761 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 25760, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod3a 25762* Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod3b 25763* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theoremrelsset 25764 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrsset 25765 For sets, the binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremidsset 25766 is equal to and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremeltrans 25767 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremdfon3 25768 A quantifier-free definition of . (Contributed by Scott Fenton, 5-Apr-2012.)

Theoremdfon4 25769 Another quantifier-free definition of . (Contributed by Scott Fenton, 4-May-2014.)

Theorembrtxpsd 25770* Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(++)

Theorembrtxpsd2 25771* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
(++)

Theorembrtxpsd3 25772* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
(++)

Theoremrelbigcup 25773 The relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Theorembrbigcup 25774 Binary relationship over . (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfbigcup2 25775 using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfobigcup 25776 maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfnbigcup 25777 is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremfvbigcup 25778 For sets, yields union. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix 25779 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix2 25780 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffix2 25781 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssdm 25782 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssrn 25783 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixcnv 25784 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixun 25785 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremellimits 25786 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremlimitssson 25787 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfom5b 25788 A quantifier-free definition of that does not depend on ax-inf 7622. (Note: label was changed from dfom5 7634 to dfom5b 25788 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremsscoid 25789 A condition for subset and composition with identity. (Contributed by Scott Fenton, 13-Apr-2018.)

Theoremdffun10 25790 Another potential definition of functionhood. Based on statements in http://people.math.gatech.edu/~belinfan/research/autoreas/otter/sum/fs/. (Contributed by Scott Fenton, 30-Aug-2017.)

Theoremelfuns 25791 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremelfunsg 25792 Closed form of elfuns 25791. (Contributed by Scott Fenton, 2-May-2014.)

Theorembrsingle 25793 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremelsingles 25794* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremfnsingle 25795 The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremfvsingle 25796 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.)
Singleton

Theoremdfsingles2 25797* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnelsingles 25798 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfiota3 25799 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv5 25800 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)

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