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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaonncan 6301* Transfer nncan 9286-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Theoremofmres 6302* Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6304, allowing it to be used as a function or structure argument. By ofmresval 6303, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Theoremofmresval 6303 Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremofmresex 6304 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Theoremsuppssof1 6305* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)

2.4.13  First and second members of an ordered pair

Syntaxc1st 6306 Extend the definition of a class to include the first member an ordered pair function.

Syntaxc2nd 6307 Extend the definition of a class to include the second member an ordered pair function.

Definitiondf-1st 6308 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6314 proves that it does this. For example, . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5310 and op1stb 4717). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Definitiondf-2nd 6309 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6315 proves that it does this. For example, . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5313 and op2ndb 5312). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Theorem1stval 6310 The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem2ndval 6311 The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theorem1st0 6312 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theorem2nd0 6313 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theoremop1st 6314 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop2nd 6315 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop1std 6316 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop2ndd 6317 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop1stg 6318 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremop2ndg 6319 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremot1stg 6320 Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6320, ot2ndg 6321, ot3rdg 6322.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot2ndg 6321 Extract the second member of an ordered triple. (See ot1stg 6320 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot3rdg 6322 Extract the third member of an ordered triple. (See ot1stg 6320 comment.) (Contributed by NM, 3-Apr-2015.)

Theorem1stval2 6323 Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theorem2ndval2 6324 Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

Theoremfo1st 6325 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo2nd 6326 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf1stres 6327 Mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf2ndres 6328 Mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo1stres 6329 Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theoremfo2ndres 6330 Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)

Theorem1st2val 6331* Value of an alternate definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem2nd2val 6332* Value of an alternate definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theorem1stcof 6333 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Theorem2ndcof 6334 Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)

Theoremxp1st 6335 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremxp2nd 6336 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremelxp6 6337 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5316. (Contributed by NM, 9-Oct-2004.)

Theoremelxp7 6338 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5316. (Contributed by NM, 19-Aug-2006.)

Theoremdifxp 6339 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp1 6340 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremdifxp2 6341 Difference law for cross product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.)

Theoremeqopi 6342 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theoremxp2 6343* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Theoremunielxp 6344 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Theorem1st2nd2 6345 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theorem1st2ndb 6346 Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)

Theoremxpopth 6347 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 6348 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 6349 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 6350* Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theorem2nd1st 6351 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 6352 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 6353 The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem2ndrn 6354 The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.)

Theorem1st2ndbr 6355 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 6356* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 6357* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremsbcopeq1a 6358 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3131 that avoids the existential quantifiers of copsexg 4402). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcsbopeq1a 6359 Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 3219). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfopab2 6360* A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab3s 6361* A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab3 6362* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Theoremdfoprab4 6363* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab4f 6364* Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfxp3 6365* Define the cross product of three classes. Compare df-xp 4843. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)

Theoremcopsex2gb 6366* Implicit substitution inference for ordered pairs. Compare copsex2ga 6367. (Contributed by NM, 12-Mar-2014.)

Theoremcopsex2ga 6367* Implicit substitution inference for ordered pairs. Compare copsex2g 4404. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremelopaba 6368* Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremexopxfr 6369* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremexopxfr2 6370* Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)

Theoremelopabi 6371* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Theoremeloprabi 6372* A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremmpt2mptsx 6373* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremmpt2mpts 6374* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

Theoremdmmpt2ssx 6375* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2x 6376* Functionality, domain and codomain of a class given by the "maps to" notation, where is not constant but depends on . (Contributed by NM, 29-Dec-2014.)

Theoremfmpt2 6377* Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2 6378* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpt2i 6379* Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremdmmpt2 6380* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)

Theoremmpt2exxg 6381* Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremmpt2exg 6382* Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremmpt2exga 6383* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.)

Theoremmpt2ex 6384* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.)

Theorembropopvvv 6385* If a binary relation holds for the result of an operation which is a result of an operation, the involved classes are sets. (Contributed by Alexander van der Vekens, 12-Dec-2017.)

Theoremmpt20 6386 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremovmptss 6387* If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremrelmpt2opab 6388* Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpt2co 6389* Composition of two functions. Variation of fmptco 5860 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremoprabco 6390* Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremoprab2co 6391* Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Theoremdf1st2 6392* An alternate possible definition of the function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdf2nd2 6393* An alternate possible definition of the function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theorem1stconst 6394 The mapping of a restriction of the function to a constant function. (Contributed by NM, 14-Dec-2008.)

Theorem2ndconst 6395 The mapping of a restriction of the function to a converse constant function. (Contributed by NM, 27-Mar-2008.)

Theoremdfmpt2 6396* Alternate definition for the "maps to" notation df-mpt2 6045 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcurry1 6397* Composition with turns any binary operation with a constant first operand into a function of the second operand only. This transformation is called "currying." (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremcurry1val 6398 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcurry1f 6399 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)

Theoremcurry2 6400* Composition with turns any binary operation with a constant second operand into a function of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)

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