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Theorem List for Intuitionistic Logic Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfuncnvcnv 4901 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 Fun  Fun  `' `'
 
Theoremfuncnv2 4902* A simpler equivalence for single-rooted (see funcnv 4903). (Contributed by NM, 9-Aug-2004.)
 Fun  `'
 
Theoremfuncnv 4903* The converse of a class is a function iff the class is single-rooted, which means that for any in the range of there is at most one such that . Definition of single-rooted in [Enderton] p. 43. See funcnv2 4902 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 Fun  `'  ran
 
Theoremfuncnv3 4904* A condition showing a class is single-rooted. (See funcnv 4903). (Contributed by NM, 26-May-2006.)
 Fun  `'  ran  dom
 
Theoremfuncnveq 4905* Another way of expressing that a class is single-rooted. Counterpart to dffun2 4855. (Contributed by Jim Kingdon, 24-Dec-2018.)
 Fun  `'
 
Theoremfun2cnv 4906* The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
 Fun  `' `'
 
Theoremsvrelfun 4907 A single-valued relation is a function. (See fun2cnv 4906 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 Fun  Rel  Fun  `' `'
 
Theoremfncnv 4908* Single-rootedness (see funcnv 4903) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
 `' R  i^i  X.  Fn  R
 
Theoremfun11 4909* Two ways of stating that is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
 Fun  `' `'  Fun  `'
 
Theoremfununi 4910* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
 Fun  C_  C_  Fun  U.
 
Theoremfuncnvuni 4911* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 4903 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
 Fun  `'  C_  C_  Fun  `' U.
 
Theoremfun11uni 4912* The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
 Fun  Fun  `'  C_  C_  Fun  U. 
 Fun  `' U.
 
Theoremfunin 4913 The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 Fun  F  Fun  F  i^i  G
 
Theoremfunres11 4914 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
 Fun  `' F  Fun  `' F  |`
 
Theoremfuncnvres 4915 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
 Fun  `' F  `' F  |`  `' F  |`  F "
 
Theoremcnvresid 4916 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 `'  _I  |`  _I  |`
 
Theoremfuncnvres2 4917 The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
 Fun  F  `' `' F  |`  F  |`  `' F "
 
Theoremfunimacnv 4918 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
 Fun  F  F " `' F "  i^i  ran 
 F
 
Theoremfunimass1 4919 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
 Fun  F  C_  ran  F  `' F "  C_  C_  F "
 
Theoremfunimass2 4920 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
 Fun  F  C_  `' F "  F "  C_
 
Theoremimadiflem 4921 One direction of imadif 4922. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 F "  \  F
 "  C_  F "  \
 
Theoremimadif 4922 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
 Fun  `' F  F "  \  F
 "  \  F "
 
Theoremimainlem 4923 One direction of imain 4924. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 F "  i^i  C_  F "  i^i  F
 "
 
Theoremimain 4924 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
 Fun  `' F  F "  i^i  F
 "  i^i  F "
 
Theoremfunimaexglem 4925 Lemma for funimaexg 4926. It constitutes the interesting part of funimaexg 4926, in which 
C_  dom . (Contributed by Jim Kingdon, 27-Dec-2018.)
 Fun  C  C_ 
 dom  " 
 _V
 
Theoremfunimaexg 4926 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
 Fun  C  "  _V
 
Theoremfunimaex 4927 The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
 _V   =>     Fun  "  _V
 
Theoremisarep1 4928* Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by  , i.e. the class  { <. ,  >.  |  } ". If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 b  { <. ,  >.  |  } "  b
 
Theoremisarep2 4929* Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature " i, i, i => o => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 4927. (Contributed by NM, 26-Oct-2006.)
 _V   &       =>     { <. ,  >.  |  } "
 
Theoremfneq1 4930 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 F  G  F  Fn  G  Fn
 
Theoremfneq2 4931 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 F  Fn  F  Fn
 
Theoremfneq1d 4932 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 F  G   =>     F  Fn  G  Fn
 
Theoremfneq2d 4933 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>     F  Fn  F  Fn
 
Theoremfneq12d 4934 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
 F  G   &       =>     F  Fn  G  Fn
 
Theoremfneq12 4935 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 F  G  F  Fn  G  Fn
 
Theoremfneq1i 4936 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 F  G   =>     F  Fn  G  Fn
 
Theoremfneq2i 4937 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
   =>     F  Fn  F  Fn
 
Theoremnffn 4938 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
 F/_ F   &     F/_   =>     F/  F  Fn
 
Theoremfnfun 4939 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 F  Fn  Fun  F
 
Theoremfnrel 4940 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 F  Fn  Rel  F
 
Theoremfndm 4941 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 F  Fn  dom  F
 
Theoremfunfni 4942 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
 Fun  F  dom  F    =>     F  Fn
 
Theoremfndmu 4943 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 F  Fn  F  Fn
 
Theoremfnbr 4944 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
 F  Fn  F C
 
Theoremfnop 4945 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
 F  Fn  <. ,  C >.  F
 
Theoremfneu 4946* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F  Fn  F
 
Theoremfneu2 4947* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
 F  Fn  <. ,  >.  F
 
Theoremfnun 4948 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 F  Fn  G  Fn  i^i  (/)  F  u.  G  Fn  u.
 
Theoremfnunsn 4949 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 X  _V   &     Y  _V   &     F  Fn  D   &     G  F  u.  { <. X ,  Y >. }   &     E  D  u.  { X }   &     X  D   =>     G  Fn  E
 
Theoremfnco 4950 Composition of two functions. (Contributed by NM, 22-May-2006.)
 F  Fn  G  Fn 
 ran  G  C_  F  o.  G  Fn
 
Theoremfnresdm 4951 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 F  Fn  F  |`  F
 
Theoremfnresdisj 4952 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
 F  Fn  i^i  (/)  F  |`  (/)
 
Theorem2elresin 4953 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
 F  Fn  G  Fn  <. ,  >.  F  <. ,  >.  G  <. ,  >.  F  |`  i^i  <. ,  >.  G  |`  i^i
 
Theoremfnssresb 4954 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
 F  Fn  F  |`  Fn  C_
 
Theoremfnssres 4955 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
 F  Fn  C_  F  |`  Fn
 
Theoremfnresin1 4956 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 F  Fn  F  |` 
 i^i  Fn  i^i
 
Theoremfnresin2 4957 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 F  Fn  F  |` 
 i^i  Fn  i^i
 
Theoremfnres 4958* An equivalence for functionality of a restriction. Compare dffun8 4872. (Contributed by Mario Carneiro, 20-May-2015.)
 F  |`  Fn  F
 
Theoremfnresi 4959 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 _I  |`  Fn
 
Theoremfnima 4960 The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F  Fn  F "  ran  F
 
Theoremfn0 4961 A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F  Fn  (/)  F  (/)
 
Theoremfnimadisj 4962 A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
 F  Fn  i^i  C  (/)  F " C  (/)
 
Theoremfnimaeq0 4963 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 F  Fn  C_  F
 "  (/)  (/)
 
Theoremdfmpt3 4964 Alternate definition for the "maps to" notation df-mpt 3811. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |->  U_  { }  X.  { }
 
Theoremfnopabg 4965* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 F  { <. ,  >.  |  }   =>     F  Fn
 
Theoremfnopab 4966* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
   &     F  { <. ,  >.  |  }   =>     F  Fn
 
Theoremmptfng 4967* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
 F  |->    =>     _V  F  Fn
 
Theoremfnmpt 4968* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
 F  |->    =>     V  F  Fn
 
Theoremmpt0 4969 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 (/)  |->  (/)
 
Theoremfnmpti 4970* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 _V   &     F  |->    =>     F  Fn
 
Theoremdmmpti 4971* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
 _V   &     F  |->    =>     dom 
 F
 
Theoremmptun 4972 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
 u.  |->  C  |->  C  u.  |->  C
 
Theoremfeq1 4973 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 F  G  F : -->  G : -->
 
Theoremfeq2 4974 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 F : --> C  F : --> C
 
Theoremfeq3 4975 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 F : C -->  F : C -->
 
Theoremfeq23 4976 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 C  D  F :
 -->  F : C --> D
 
Theoremfeq1d 4977 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 F  G   =>     F : -->  G :
 -->
 
Theoremfeq2d 4978 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>     F : --> C  F :
 --> C
 
Theoremfeq12d 4979 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 F  G   &       =>     F : --> C  G :
 --> C
 
Theoremfeq123d 4980 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 F  G   &       &     C  D   =>     F : --> C  G :
 --> D
 
Theoremfeq123 4981 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 F  G  C  D  F : -->  G : C --> D
 
Theoremfeq1i 4982 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 F  G   =>     F : -->  G : -->
 
Theoremfeq2i 4983 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
   =>     F : --> C  F : --> C
 
Theoremfeq23i 4984 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 C   &     D   =>     F : -->  F : C --> D
 
Theoremfeq23d 4985 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 C   &     D   =>     F : -->  F : C
 --> D
 
Theoremnff 4986 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 F/_ F   &     F/_   &     F/_   =>    
 F/  F :
 -->
 
Theoremsbcfng 4987* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 X  V  [. X  ]. F  Fn  [_ X  ]_ F  Fn  [_ X  ]_
 
Theoremsbcfg 4988* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 X  V  [. X  ]. F : --> 
 [_ X  ]_ F : [_ X  ]_ --> [_ X  ]_
 
Theoremffn 4989 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 F : -->  F  Fn
 
Theoremdffn2 4990 Any function is a mapping into  _V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F  Fn  F : --> _V
 
Theoremffun 4991 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 F : -->  Fun  F
 
Theoremfrel 4992 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 F : -->  Rel  F
 
Theoremfdm 4993 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 F : -->  dom  F
 
Theoremfdmi 4994 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 F : -->   =>     dom  F
 
Theoremfrn 4995 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 F : -->  ran  F  C_
 
Theoremdffn3 4996 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 F  Fn  F : --> ran  F
 
Theoremfss 4997 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F :
 -->  C_  C  F :
 --> C
 
Theoremfssd 4998 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 F : -->   &     C_  C   =>     F : --> C
 
Theoremfco 4999 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 F :
 --> C  G :
 -->  F  o.  G :
 --> C
 
Theoremfco2 5000 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 F  |`  : --> C  G : -->  F  o.  G :
 --> C
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