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Theorem List for Intuitionistic Logic Explorer - 6001-6100   *Has distinct variable group(s)
TypeLabelDescription
Statement

2.6.13  Functions (continued)

TheoremresfunexgALT 6001 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5634 but requires ax-pow 4093 and ax-un 4350. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcofunexg 6002 Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)

Theoremcofunex2g 6003 Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)

TheoremfnexALT 6004 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5202. This version of fnex 5635 uses ax-pow 4093 and ax-un 4350, whereas fnex 5635 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremfunrnex 6005 If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5636. (Contributed by NM, 11-Nov-1995.)

Theoremfornex 6006 If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)

Theoremf1dmex 6007 If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)

Theoremabrexex 6008* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in the class expression substituted for , which can be thought of as . This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5638, funex 5636, fnex 5635, resfunexg 5634, and funimaexg 5202. See also abrexex2 6015. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Theoremabrexexg 6009* Existence of a class abstraction of existentially restricted sets. is normally a free-variable parameter in . The antecedent assures us that is a set. (Contributed by NM, 3-Nov-2003.)

Theoremiunexg 6010* The existence of an indexed union. is normally a free-variable parameter in . (Contributed by NM, 23-Mar-2006.)

Theoremabrexex2g 6011* Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremopabex3d 6012* Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)

Theoremopabex3 6013* Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremiunex 6014* The existence of an indexed union. is normally a free-variable parameter in the class expression substituted for , which can be read informally as . (Contributed by NM, 13-Oct-2003.)

Theoremabrexex2 6015* Existence of an existentially restricted class abstraction. is normally has free-variable parameters and . See also abrexex 6008. (Contributed by NM, 12-Sep-2004.)

Theoremabexssex 6016* Existence of a class abstraction with an existentially quantified expression. Both and can be free in . (Contributed by NM, 29-Jul-2006.)

Theoremabexex 6017* A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)

Theoremoprabexd 6018* Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremoprabex 6019* Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)

Theoremoprabex3 6020* Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)

Theoremoprabrexex2 6021* Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)

Theoremab2rexex 6022* Existence of a class abstraction of existentially restricted sets. Variables and are normally free-variable parameters in the class expression substituted for , which can be thought of as . See comments for abrexex 6008. (Contributed by NM, 20-Sep-2011.)

Theoremab2rexex2 6023* Existence of an existentially restricted class abstraction. normally has free-variable parameters , , and . Compare abrexex2 6015. (Contributed by NM, 20-Sep-2011.)

TheoremxpexgALT 6024 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4648 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremoffval3 6025* General value of with no assumptions on functionality of and . (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremoffres 6026 Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Theoremofmres 6027* Equivalent expressions for a restriction of the function operation map. Unlike which is a proper class, can be a set by ofmresex 6028, allowing it to be used as a function or structure argument. By ofmresval 5986, the restricted operation map values are the same as the original values, allowing theorems for to be reused. (Contributed by NM, 20-Oct-2014.)

Theoremofmresex 6028 Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

2.6.14  First and second members of an ordered pair

Syntaxc1st 6029 Extend the definition of a class to include the first member an ordered pair function.

Syntaxc2nd 6030 Extend the definition of a class to include the second member an ordered pair function.

Definitiondf-1st 6031 Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6037 proves that it does this. For example, ( 3 , 4 ) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5015 and op1stb 4394). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Definitiondf-2nd 6032 Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6038 proves that it does this. For example, 3 , 4 ) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5018 and op2ndb 5017). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Theorem1stvalg 6033 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.)

Theorem2ndvalg 6034 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 6035 The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theorem2nd0 6036 The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)

Theoremop1st 6037 Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop2nd 6038 Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Theoremop1std 6039 Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop2ndd 6040 Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)

Theoremop1stg 6041 Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremop2ndg 6042 Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Theoremot1stg 6043 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 6043, ot2ndg 6044, ot3rdgg 6045.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot2ndg 6044 Extract the second member of an ordered triple. (See ot1stg 6043 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Theoremot3rdgg 6045 Extract the third member of an ordered triple. (See ot1stg 6043 comment.) (Contributed by NM, 3-Apr-2015.)

Theorem1stval2 6046 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 6047 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 6048 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremfo2nd 6049 The function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremf1stres 6050 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 6051 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.)

Theoremfo1stresm 6052* Onto mapping of a restriction of the (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Theoremfo2ndresm 6053* Onto mapping of a restriction of the (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)

Theorem1stcof 6054 Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)

Theorem2ndcof 6055 Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.)

Theoremxp1st 6056 Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremxp2nd 6057 Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem1stexg 6058 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Theorem2ndexg 6059 Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Theoremelxp6 6060 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5021. (Contributed by NM, 9-Oct-2004.)

Theoremelxp7 6061 Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5021. (Contributed by NM, 19-Aug-2006.)

Theoremoprssdmm 6062* Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)

Theoremeqopi 6063 Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Theoremxp2 6064* Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Theoremunielxp 6065 The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.)

Theorem1st2nd2 6066 Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.)

Theoremxpopth 6067 An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Theoremeqop 6068 Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Theoremeqop2 6069 Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Theoremop1steq 6070* 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 6071 Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Theorem1st2nd 6072 Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)

Theorem1stdm 6073 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 6074 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 6075 Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)

Theoremreleldm2 6076* Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremreldm 6077* An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.)

Theoremsbcopeq1a 6078 Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2913 that avoids the existential quantifiers of copsexg 4161). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremcsbopeq1a 6079 Equality theorem for substitution of a class for an ordered pair in (analog of csbeq1a 3007). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfopab2 6080* 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 6081* 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 6082* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Theoremdfoprab4 6083* Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Theoremdfoprab4f 6084* 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 6085* Define the cross product of three classes. Compare df-xp 4540. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)

Theoremelopabi 6086* A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Theoremeloprabi 6087* 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.)

Theoremmpomptsx 6088* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremmpompts 6089* Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

Theoremdmmpossx 6090* The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Theoremfmpox 6091* 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.)

Theoremfmpo 6092* Functionality, domain and range of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Theoremfnmpo 6093* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Theoremmpofvex 6094* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremfnmpoi 6095* Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Theoremdmmpo 6096* Domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)

Theoremmpofvexi 6097* Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.)

Theoremovmpoelrn 6098* An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.)

Theoremdmmpoga 6099* Domain of an operation given by the maps-to notation, closed form of dmmpo 6096. (Contributed by Alexander van der Vekens, 10-Feb-2019.)

Theoremdmmpog 6100* Domain of an operation given by the maps-to notation, closed form of dmmpo 6096. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.)

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