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

Theoremtxpss3v 25601 A tail cross product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremtxprel 25602 A tail cross product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp 25603 Characterize a trinary relationship over a tail cross product. Together with txpss3v 25601, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrtxp2 25604* 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 25605 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theorempprodcnveq 25606 A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod pprod

Theorempprodss4v 25607 The parallel product is a subclass of . (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod 25608 Characterize a quatary relationship over a tail cross product. Together with pprodss4v 25607, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
pprod

Theorembrpprod3a 25609* 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 25610* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theoremrelsset 25611 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrsset 25612 For sets, the binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremidsset 25613 is equal to and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremeltrans 25614 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremdfon3 25615 A quantifier-free definition of . (Contributed by Scott Fenton, 5-Apr-2012.)

Theoremdfon4 25616 Another quantifier-free definition of . (Contributed by Scott Fenton, 4-May-2014.)

Theorembrtxpsd 25617* 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 25618* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
(++)

Theorembrtxpsd3 25619* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
(++)

Theoremrelbigcup 25620 The relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Theorembrbigcup 25621 Binary relationship over . (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfbigcup2 25622 using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfobigcup 25623 maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfnbigcup 25624 is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremfvbigcup 25625 For sets, yields union. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix 25626 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix2 25627 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffix2 25628 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssdm 25629 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssrn 25630 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixcnv 25631 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixun 25632 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremellimits 25633 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremlimitssson 25634 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfom5b 25635 A quantifier-free definition of that does not depend on ax-inf 7540. (Note: label was changed from dfom5 7552 to dfom5b 25635 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffun10 25636 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 25637 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremelfunsg 25638 Closed form of elfuns 25637. (Contributed by Scott Fenton, 2-May-2014.)

Theorembrsingle 25639 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremelsingles 25640* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremfnsingle 25641 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 25642 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremdfsingles2 25643* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnelsingles 25644 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfiota3 25645 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv5 25646 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremunisnif 25647 Express union of singleton in terms of . (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrimage 25648 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrimageg 25649 Closed form of brimage 25648. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfunimage 25650 Image is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfnimage 25651* 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 25652* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfvimage 25653 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrcart 25654 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cart

Theorembrdomain 25655 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Domain

Theorembrrange 25656 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Range

Theorembrdomaing 25657 Closed form of brdomain 25655. (Contributed by Scott Fenton, 2-May-2014.)
Domain

Theorembrrangeg 25658 Closed form of brrange 25656. (Contributed by Scott Fenton, 3-May-2014.)
Range

Theorembrimg 25659 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Img

Theorembrapply 25660 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Apply

Theorembrcup 25661 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cup

Theorembrcap 25662 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cap

Theorembrsuccf 25663 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Succ

Theoremfunpartlem 25664* Lemma for funpartfun 25665. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
Image Singleton

Theoremfunpartfun 25665 The functional part of is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartss 25666 The functional part of is a subset of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartfv 25667 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 25668 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 25669 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

Theorembrfullfun 25670 A binary relationship form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
FullFun

Theorembrrestrict 25671 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Restrict

Theoremdfrdg4 25672 A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Domain Domain Apply Img FullFun Apply pprod Succ

Theoremtfrqfree 25673* Calculate a quantifier-free version of the function from tfr1 6608 through tfr3 6610. (Contributed by Scott Fenton, 29-Apr-2014.)
Domain Domain Apply FullFun Restrict

19.7.38  Alternate ordered pairs

Syntaxcaltop 25674 Declare the syntax for an alternate ordered pair.

Syntaxcaltxp 25675 Declare the syntax for an alternate cross product.

Definitiondf-altop 25676 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 25687), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)

Definitiondf-altxp 25677* Define cross products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopex 25678 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthsn 25679 Two alternate ordered pairs are equal iff the singletons of their respective elements are equal. Note that this holds regardless of sethood of any of the elements. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremaltopeq12 25680 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq1 25681 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq2 25682 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopth1 25683 Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopth2 25684 Equality of the second members of equal alternate ordered pairs, which holds regardless of the first members' sethood. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthg 25685 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthbg 25686 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopth 25687 The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that and are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4390), requires to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopthb 25688 Alternate ordered pair theorem with different sethood requirements. See altopth 25687 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthc 25689 Alternate ordered pair theorem with different sethood requirements. See altopth 25687 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthd 25690 Alternate ordered pair theorem with different sethood requirements. See altopth 25687 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltxpeq1 25691 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpeq2 25692 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremelaltxp 25693* Membership in alternate cross products. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopelaltxp 25694 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4862, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltxpsspw 25695 An inclusion rule for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpexg 25696 The alternate cross product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremrankaltopb 25697 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremnfaltop 25698 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

Theoremsbcaltop 25699* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

19.7.39  Tarskian geometry

Syntaxcee 25700 Declare the syntax for the Euclidean space generator.

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