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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembrpprod3a 23601* 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 23602* Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.)
pprod

Theoremrelsset 23603 The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theorembrsset 23604 For sets, the binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremidsset 23605 is equal to and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremeltrans 23606 Membership in the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)

Theoremdfon3 23607 A quantifier-free definition of . (Contributed by Scott Fenton, 5-Apr-2012.)

Theoremdfon4 23608 Another quantifier-free definition of . (Contributed by Scott Fenton, 4-May-2014.)

Theorembrtxpsd 23609* 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 23610* Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.)
(++)

Theorembrtxpsd3 23611* A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.)
(++)

Theoremrelbigcup 23612 The relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)

Theorembrbigcup 23613 Binary relationship over . (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfbigcup2 23614 using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfobigcup 23615 maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfnbigcup 23616 is a function over the universal class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremfvbigcup 23617 For sets, yields union. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix 23618 Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfix2 23619 Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdffix2 23620 The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssdm 23621 The fixpoints of a class are a subset of its domain. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixssrn 23622 The fixpoints of a class are a subset of its range. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixcnv 23623 The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremfixun 23624 The fixpoint operator distributes over union. (Contributed by Scott Fenton, 16-Apr-2012.)

Theoremellimits 23625 Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremlimitssson 23626 The class of all limit ordinals is a subclass of the class of all ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremdfom5b 23627 A quantifier-free definition of that does not depend on ax-inf 7223. (Note: label was changed from dfom5 7235 to dfom5b 23627 to prevent naming conflict. NM 12-Feb-2013) (Contributed by Scott Fenton, 11-Apr-2012.)

Theoremelfuns 23628 Membership in the class of all functions. (Contributed by Scott Fenton, 18-Feb-2013.)

Theoremelfunsg 23629 Closed form of elfuns 23628. (Contributed by Scott Fenton, 2-May-2014.)

Theorembrsingle 23630 The binary relationship form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremelsingles 23631* Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremfnsingle 23632 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 23633 The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Singleton

Theoremdfsingles2 23634* Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremsnelsingles 23635 A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfiota3 23636 A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdffv4 23637 Another quantifier free definition of function value. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremunisnif 23638 Express union of singleton in terms of . (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theorembrimage 23639 Binary relationship form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrimageg 23640 Closed form of brimage 23639. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfunimage 23641 Image is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfnimage 23642* 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 23643* The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theoremfvimage 23644 The value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Image

Theorembrcart 23645 Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cart

Theorembrdomain 23646 The binary relationship form of the domain function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Domain

Theorembrrange 23647 The binary relationship form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Range

Theorembrdomaing 23648 Closed form of brdomain 23646. (Contributed by Scott Fenton, 2-May-2014.)
Domain

Theorembrrangeg 23649 Closed form of brrange 23647. (Contributed by Scott Fenton, 3-May-2014.)
Range

Theorembrimg 23650 The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Img

Theorembrapply 23651 The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Apply

Theorembrcup 23652 Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cup

Theorembrcap 23653 Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Cap

Theorembrsuccf 23654 Binary relationship form of the Succ function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Succ

Theoremfunpartfun 23655 The functional part of is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartss 23656 The functional part of is a subset of . (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Funpart

Theoremfunpartfv 23657 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 23658 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 23659 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 23660 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 23661 The binary relationship form of the Restrict function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Restrict

Theoremdfrdg4 23662 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 23663* Calculate a quantifier-free version of the function from tfr1 6299 through tfr3 6301. (Contributed by Scott Fenton, 29-Apr-2014.)
Domain Domain Apply FullFun Restrict

16.6.29  Alternate ordered pairs

Syntaxcaltop 23664 Declare the syntax for an alternate ordered pair.

Syntaxcaltxp 23665 Declare the syntax for an alternate cross product.

Definitiondf-altop 23666 An alternative definition of ordered pairs. This definition removes a hypothesis from its defining theorem (see altopth 23677), making it more convenient in some circumstances. (Contributed by Scott Fenton, 22-Mar-2012.)

Definitiondf-altxp 23667* Define cross products of alternative ordered pairs. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopex 23668 Alternative ordered pairs always exist. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthsn 23669 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 23670 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq1 23671 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopeq2 23672 Equality for alternate ordered pairs. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopth1 23673 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 23674 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 23675 Alternate ordered pair theorem. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltopthbg 23676 Alternate ordered pair theorem. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopth 23677 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 4138), requires to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopthb 23678 Alternate ordered pair theorem with different sethood requirements. See altopth 23677 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthc 23679 Alternate ordered pair theorem with different sethood requirements. See altopth 23677 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltopthd 23680 Alternate ordered pair theorem with different sethood requirements. See altopth 23677 for more comments. (Contributed by Scott Fenton, 14-Apr-2012.)

Theoremaltxpeq1 23681 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpeq2 23682 Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremelaltxp 23683* Membership in alternate cross products. (Contributed by Scott Fenton, 23-Mar-2012.)

Theoremaltopelaltxp 23684 Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4626, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)

Theoremaltxpsspw 23685 An inclusion rule for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremaltxpexg 23686 The alternate cross product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)

Theoremrankaltopb 23687 Compute the rank of an alternate ordered pair. (Contributed by Scott Fenton, 18-Dec-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Theoremnfaltop 23688 Bound-variable hypothesis builder for alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

Theoremsbcaltop 23689* Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

16.6.30  Tarskian geometry

Syntaxcee 23690 Declare the syntax for the Euclidean space generator.

Syntaxcbtwn 23691 Declare the syntax for the Euclidean betweenness predicate.

Syntaxccgr 23692 Declare the syntax for the Euclidean congruence predicate.
Cgr

Definitiondf-ee 23693 Define the Euclidean space generator. For details, see elee 23696. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-btwn 23694* Define the Euclidean betweenness predicate. For details, see brbtwn 23701. (Contributed by Scott Fenton, 3-Jun-2013.)

Definitiondf-cgr 23695* Define the Euclidean congruence predicate. For details, see brcgr 23702. (Contributed by Scott Fenton, 3-Jun-2013.)
Cgr

Theoremelee 23696 Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)

Theoremmptelee 23697* A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)

Theoremeleenn 23698 If is in , then is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeleei 23699 The forward direction of elee 23696. (Contributed by Scott Fenton, 1-Jul-2013.)

Theoremeedimeq 23700 A point belongs to at most one Euclidean space. (Contributed by Scott Fenton, 1-Jul-2013.)

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