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Theorem List for Intuitionistic Logic Explorer - 5001-5100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnvresid 5001 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 |-  `' (  _I  |`  A )  =  (  _I  |`  A )
 
Theoremfuncnvres2 5002 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  |`  A )  =  ( F  |`  ( `' F " A ) ) )
 
Theoremfunimacnv 5003 The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
 |-  ( Fun  F  ->  ( F " ( `' F " A ) )  =  ( A  i^i  ran  F )
 )
 
Theoremfunimass1 5004 A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
 |-  ( ( Fun  F  /\  A  C_  ran  F ) 
 ->  ( ( `' F " A )  C_  B  ->  A  C_  ( F " B ) ) )
 
Theoremfunimass2 5005 A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
 |-  ( ( Fun  F  /\  A  C_  ( `' F " B ) ) 
 ->  ( F " A )  C_  B )
 
Theoremimadiflem 5006 One direction of imadif 5007. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( ( F " A )  \  ( F
 " B ) ) 
 C_  ( F "
 ( A  \  B ) )
 
Theoremimadif 5007 The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
 |-  ( Fun  `' F  ->  ( F " ( A  \  B ) )  =  ( ( F
 " A )  \  ( F " B ) ) )
 
Theoremimainlem 5008 One direction of imain 5009. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( F " ( A  i^i  B ) ) 
 C_  ( ( F
 " A )  i^i  ( F " B ) )
 
Theoremimain 5009 The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
 |-  ( Fun  `' F  ->  ( F " ( A  i^i  B ) )  =  ( ( F
 " A )  i^i  ( F " B ) ) )
 
Theoremfunimaexglem 5010 Lemma for funimaexg 5011. It constitutes the interesting part of funimaexg 5011, in which  B 
C_  dom  A. (Contributed by Jim Kingdon, 27-Dec-2018.)
 |-  ( ( Fun  A  /\  B  e.  C  /\  B  C_  dom  A )  ->  ( A " B )  e.  _V )
 
Theoremfunimaexg 5011 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  A  /\  B  e.  C ) 
 ->  ( A " B )  e.  _V )
 
Theoremfunimaex 5012 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.)
 |-  B  e.  _V   =>    |-  ( Fun  A  ->  ( A " B )  e.  _V )
 
Theoremisarep1 5013* 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  ph ( x ,  y ) i.e. the class  ( {
<. x ,  y >.  |  ph } " A
). 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  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
 
Theoremisarep2 5014* 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 5012. (Contributed by NM, 26-Oct-2006.)
 |-  A  e.  _V   &    |-  A. x  e.  A  A. y A. z ( ( ph  /\ 
 [ z  /  y ] ph )  ->  y  =  z )   =>    |- 
 E. w  w  =  ( { <. x ,  y >.  |  ph } " A )
 
Theoremfneq1 5015 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5016 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5017 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2d 5018 Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq12d 5019 Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  B ) )
 
Theoremfneq12 5020 Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  ( ( F  =  G  /\  A  =  B )  ->  ( F  Fn  A 
 <->  G  Fn  B ) )
 
Theoremfneq1i 5021 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5022 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5023 Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/ x  F  Fn  A
 
Theoremfnfun 5024 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5025 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5026 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5027 Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
 |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ph )   =>    |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
 
Theoremfndmu 5028 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5029 The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
 |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
 
Theoremfnop 5030 The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )
 
Theoremfneu 5031* There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y  B F y )
 
Theoremfneu2 5032* There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
 |-  ( ( F  Fn  A  /\  B  e.  A )  ->  E! y <. B ,  y >.  e.  F )
 
Theoremfnun 5033 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 |-  ( ( ( F  Fn  A  /\  G  Fn  B )  /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G )  Fn  ( A  u.  B ) )
 
Theoremfnunsn 5034 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  Y  e.  _V )   &    |-  ( ph  ->  F  Fn  D )   &    |-  G  =  ( F  u.  { <. X ,  Y >. } )   &    |-  E  =  ( D  u.  { X } )   &    |-  ( ph  ->  -.  X  e.  D )   =>    |-  ( ph  ->  G  Fn  E )
 
Theoremfnco 5035 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfnresdm 5036 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5037 A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
 |-  ( F  Fn  A  ->  ( ( A  i^i  B )  =  (/)  <->  ( F  |`  B )  =  (/) ) )
 
Theorem2elresin 5038 Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
 |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( <. x ,  y >.  e.  F  /\  <. x ,  z >.  e.  G )  <->  ( <. x ,  y >.  e.  ( F  |`  ( A  i^i  B ) )  /\  <. x ,  z >.  e.  ( G  |`  ( A  i^i  B ) ) ) ) )
 
Theoremfnssresb 5039 Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
 |-  ( F  Fn  A  ->  ( ( F  |`  B )  Fn  B  <->  B  C_  A ) )
 
Theoremfnssres 5040 Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( F  |`  B )  Fn  B )
 
Theoremfnresin1 5041 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( A  i^i  B ) )  Fn  ( A  i^i  B ) )
 
Theoremfnresin2 5042 Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
 |-  ( F  Fn  A  ->  ( F  |`  ( B  i^i  A ) )  Fn  ( B  i^i  A ) )
 
Theoremfnres 5043* An equivalence for functionality of a restriction. Compare dffun8 4957. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5044 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5045 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  A  ->  ( F " A )  =  ran  F )
 
Theoremfn0 5046 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 5047 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  A  /\  ( A  i^i  C )  =  (/) )  ->  ( F " C )  =  (/) )
 
Theoremfnimaeq0 5048 Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( ( F
 " B )  =  (/) 
 <->  B  =  (/) ) )
 
Theoremdfmpt3 5049 Alternate definition for the "maps to" notation df-mpt 3848. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
 )
 
Theoremfnopabg 5050* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
 |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }   =>    |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A )
 
Theoremfnopab 5051* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
 |-  ( x  e.  A  ->  E! y ph )   &    |-  F  =  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }   =>    |-  F  Fn  A
 
Theoremmptfng 5052* The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A )
 
Theoremfnmpt 5053* The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
 
Theoremmpt0 5054 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( x  e.  (/)  |->  A )  =  (/)
 
Theoremfnmpti 5055* Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  F  Fn  A
 
Theoremdmmpti 5056* Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  B  e.  _V   &    |-  F  =  ( x  e.  A  |->  B )   =>    |- 
 dom  F  =  A
 
Theoremmptun 5057 Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( x  e.  ( A  u.  B )  |->  C )  =  ( ( x  e.  A  |->  C )  u.  ( x  e.  B  |->  C ) )
 
Theoremfeq1 5058 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A --> B 
 <->  G : A --> B ) )
 
Theoremfeq2 5059 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A --> C 
 <->  F : B --> C ) )
 
Theoremfeq3 5060 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C --> A 
 <->  F : C --> B ) )
 
Theoremfeq23 5061 Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A
 --> B  <->  F : C --> D ) )
 
Theoremfeq1d 5062 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A --> B  <->  G : A --> B ) )
 
Theoremfeq2d 5063 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  F : B --> C ) )
 
Theoremfeq12d 5064 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  G : B --> C ) )
 
Theoremfeq123d 5065 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A --> C  <->  G : B --> D ) )
 
Theoremfeq123 5066 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D ) 
 ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremfeq1i 5067 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F : A
 --> B  <->  G : A --> B )
 
Theoremfeq2i 5068 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
 |-  A  =  B   =>    |-  ( F : A
 --> C  <->  F : B --> C )
 
Theoremfeq23i 5069 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( F : A --> B 
 <->  F : C --> D )
 
Theoremfeq23d 5070 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( F : A --> B  <->  F : C --> D ) )
 
Theoremnff 5071 Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A
 --> B
 
Theoremsbcfng 5072* Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  ( X  e.  V  ->  ( [. X  /  x ]. F  Fn  A  <->  [_ X  /  x ]_ F  Fn  [_ X  /  x ]_ A ) )
 
Theoremsbcfg 5073* Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
 |-  ( X  e.  V  ->  ( [. X  /  x ]. F : A --> B 
 <-> 
 [_ X  /  x ]_ F : [_ X  /  x ]_ A --> [_ X  /  x ]_ B ) )
 
Theoremffn 5074 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  F  Fn  A )
 
Theoremdffn2 5075 Any function is a mapping into  _V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F  Fn  A  <->  F : A --> _V )
 
Theoremffun 5076 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Fun  F )
 
Theoremfrel 5077 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Rel  F )
 
Theoremfdm 5078 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  dom  F  =  A )
 
Theoremfdmi 5079 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 |-  F : A --> B   =>    |-  dom  F  =  A
 
Theoremfrn 5080 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  ran  F  C_  B )
 
Theoremdffn3 5081 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 |-  ( F  Fn  A  <->  F : A --> ran  F )
 
Theoremfss 5082 Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F : A
 --> B  /\  B  C_  C )  ->  F : A
 --> C )
 
Theoremfssd 5083 Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfco 5084 Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( ( F : B
 --> C  /\  G : A
 --> B )  ->  ( F  o.  G ) : A --> C )
 
Theoremfco2 5085 Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
 
Theoremfssxp 5086 A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
 
Theoremfex2 5087 A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F : A
 --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
 
Theoremfunssxp 5088 Two ways of specifying a partial function from  A to  B. (Contributed by NM, 13-Nov-2007.)
 |-  ( ( Fun  F  /\  F  C_  ( A  X.  B ) )  <->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremffdm 5089 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 |-  ( F : A --> B  ->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremopelf 5090 The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( F : A
 --> B  /\  <. C ,  D >.  e.  F ) 
 ->  ( C  e.  A  /\  D  e.  B ) )
 
Theoremfun 5091 The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
 |-  ( ( ( F : A --> C  /\  G : B --> D ) 
 /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B ) --> ( C  u.  D ) )
 
Theoremfun2 5092 The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( ( F : A --> C  /\  G : B --> C ) 
 /\  ( A  i^i  B )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  B ) --> C )
 
Theoremfnfco 5093 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfssres 5094 Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
 |-  ( ( F : A
 --> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
 
Theoremfssres2 5095 Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
 |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  ->  ( F  |`  C ) : C --> B )
 
Theoremfresin 5096 An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
 |-  ( F : A --> B  ->  ( F  |`  X ) : ( A  i^i  X ) --> B )
 
Theoremresasplitss 5097 If two functions agree on their common domain, their union contains a union of three functions with pairwise disjoint domains. If we assumed the law of the excluded middle, this would be equality rather than subset. (Contributed by Jim Kingdon, 28-Dec-2018.)
 |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) )  ->  ( ( F  |`  ( A  i^i  B ) )  u.  (
 ( F  |`  ( A 
 \  B ) )  u.  ( G  |`  ( B 
 \  A ) ) ) )  C_  ( F  u.  G ) )
 
Theoremfcoi1 5098 Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  ( F  o.  (  _I  |`  A )
 )  =  F )
 
Theoremfcoi2 5099 Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
 |-  ( F : A --> B  ->  ( (  _I  |`  B )  o.  F )  =  F )
 
Theoremfeu 5100* There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
 |-  ( ( F : A
 --> B  /\  C  e.  A )  ->  E! y  e.  B  <. C ,  y >.  e.  F )
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