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Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcrange 25601 Declare the syntax for the range function.
Range

Syntaxcapply 25602 Declare the syntax for the application function.
Apply

Syntaxccup 25603 Declare the syntax for the cup function.
Cup

Syntaxccap 25604 Declare the syntax for the cap function.
Cap

Syntaxcsuccf 25605 Declare the syntax for the successor function.
Succ

Syntaxcfunpart 25606 Declare the syntax for the functional part functor.
Funpart

Syntaxcfullfn 25607 Declare the syntax for the full function functor.
FullFun

Syntaxcrestrict 25608 Declare the syntax for the restriction function.
Restrict

Definitiondf-txp 25609 Define the tail cross of two classes. Membership in this class is defined by txpss3v 25632 and brtxp 25634. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-pprod 25610 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 25638 and brpprod 25639. (Contributed by Scott Fenton, 11-Apr-2014.)
pprod

Definitiondf-sset 25611 Define the subset class. For the value, see brsset 25643. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-trans 25612 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Definitiondf-bigcup 25613 Define the Bigcup function, which, per fvbigcup 25656, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
(++)

Definitiondf-fix 25614 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Definitiondf-limits 25615 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Definitiondf-funs 25616 Define the class of all functions. See elfuns 25668 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)

Definitiondf-singleton 25617 Define the singleton function. See brsingle 25670 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton (++)

Definitiondf-singles 25618 Define the class of all singletons. See elsingles 25671 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singleton

Definitiondf-image 25619 Define the image functor. This function takes a set to a function , providing that the latter exists. See imageval 25683 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image (++)

Definitiondf-cart 25620 Define the cartesian product function. See brcart 25685 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart (++)pprod

Definitiondf-img 25621 Define the image function. See brimg 25690 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img Image Cart

Definitiondf-domain 25622 Define the domain function. See brdomain 25686 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain Image

Definitiondf-range 25623 Define the range function. See brrange 25687 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range Image

Definitiondf-cup 25624 Define the little cup function. See brcup 25692 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup (++)

Definitiondf-cap 25625 Define the little cap function. See brcap 25693 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap (++)

Definitiondf-restrict 25626 Define the restriction function. See brrestrict 25702 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict Cap Cart Range

Definitiondf-succf 25627 Define the successor function. See brsuccf 25694 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ Cup Singleton

Definitiondf-apply 25628 Define the application function. See brapply 25691 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Apply (++) Singleton Img pprod Singleton

Definitiondf-funpart 25629 Define the functional part of a class . This is the maximal part of that is a function. See funpartfun 25696 and funpartfv 25698 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart Image Singleton

Definitiondf-fullfun 25630 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

Theorembrv 25631 The binary relationship over always holds. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremtxpss3v 25632 A tail cross product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremtxprel 25633 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp 25634 Characterize a trinary relationship over a tail cross product. Together with txpss3v 25632, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp2 25635* 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 25636 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theorempprodcnveq 25637 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod pprod

Theorempprodss4v 25638 The parallel product is a subclass of . (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod 25639 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 25638, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod3a 25640* 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 25641* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theoremrelsset 25642 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrsset 25643 For sets, the binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremidsset 25644 is equal to and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremeltrans 25645 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremdfon3 25646 A quantifier-free definition of . (Contributed by Scott Fenton, 5-Apr-2012.)

Theoremdfon4 25647 Another quantifier-free definition of . (Contributed by Scott Fenton, 4-May-2014.)

Theorembrtxpsd 25648* 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 25649* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
(++)

Theorembrtxpsd3 25650* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
(++)

Theoremrelbigcup 25651 The relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Theorembrbigcup 25652 Binary relationship over . (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfbigcup2 25653 using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfobigcup 25654 maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfnbigcup 25655 is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremfvbigcup 25656 For sets, yields union. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix 25657 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix2 25658 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffix2 25659 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssdm 25660 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssrn 25661 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixcnv 25662 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixun 25663 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremellimits 25664 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremlimitssson 25665 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfom5b 25666 A quantifier-free definition of that does not depend on ax-inf 7549. (Note: label was changed from dfom5 7561 to dfom5b 25666 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffun10 25667 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 25668 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremelfunsg 25669 Closed form of elfuns 25668. (Contributed by Scott Fenton, 2-May-2014.)

Theorembrsingle 25670 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremelsingles 25671* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremfnsingle 25672 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 25673 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremdfsingles2 25674* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnelsingles 25675 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfiota3 25676 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv5 25677 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremunisnif 25678 Express union of singleton in terms of . (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrimage 25679 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrimageg 25680 Closed form of brimage 25679. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfunimage 25681 Image is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfnimage 25682* Image is a function over the set-like portion of . (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremimageval 25683* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfvimage 25684 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrcart 25685 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cart

Theorembrdomain 25686 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Domain

Theorembrrange 25687 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Range

Theorembrdomaing 25688 Closed form of brdomain 25686. (Contributed by Scott Fenton, 2-May-2014.)
Domain

Theorembrrangeg 25689 Closed form of brrange 25687. (Contributed by Scott Fenton, 3-May-2014.)
Range

Theorembrimg 25690 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Img

Theorembrapply 25691 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Apply

Theorembrcup 25692 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cup

Theorembrcap 25693 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cap

Theorembrsuccf 25694 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Succ

Theoremfunpartlem 25695* Lemma for funpartfun 25696. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
Image Singleton

Theoremfunpartfun 25696 The functional part of is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartss 25697 The functional part of is a subset of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartfv 25698 The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfullfunfnv 25699 The full functional part of is a function over . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
FullFun

Theoremfullfunfv 25700 The function value of the full function of agrees with . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
FullFun

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