HomeHome Intuitionistic Logic Explorer
Theorem List (p. 53 of 134)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfunimass2 5201 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 5202 One direction of imadif 5203. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
 |-  ( ( F " A )  \  ( F
 " B ) ) 
 C_  ( F "
 ( A  \  B ) )
 
Theoremimadif 5203 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 5204 One direction of imain 5205. 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 5205 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 5206 Lemma for funimaexg 5207. It constitutes the interesting part of funimaexg 5207, 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 5207 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 5208 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 5209* 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 5210* 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 5208. (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 5211 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5212 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5213 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 5214 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 5215 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 5216 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 5217 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5218 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5219 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 5220 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5221 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5222 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5223 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 5224 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5225 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 5226 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 5227* 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 5228* 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 5229 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 5230 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 5231 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 5232 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5233 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 5234 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 5235 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 5236 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 5237 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 5238 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 5239* An equivalence for functionality of a restriction. Compare dffun8 5151. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5240 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5241 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 5242 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 5243 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 5244 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 5245 Alternate definition for the maps-to notation df-mpt 3991. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
 )
 
Theoremfnopabg 5246* 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 5247* 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 5248* 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 5249* 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 5250 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( x  e.  (/)  |->  A )  =  (/)
 
Theoremfnmpti 5251* 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 5252* 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
 
Theoremdmmptd 5253* The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  A  =  ( x  e.  B  |->  C )   &    |-  ( ( ph  /\  x  e.  B )  ->  C  e.  V )   =>    |-  ( ph  ->  dom  A  =  B )
 
Theoremmptun 5254 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 5255 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A --> B 
 <->  G : A --> B ) )
 
Theoremfeq2 5256 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A --> C 
 <->  F : B --> C ) )
 
Theoremfeq3 5257 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C --> A 
 <->  F : C --> B ) )
 
Theoremfeq23 5258 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 5259 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A --> B  <->  G : A --> B ) )
 
Theoremfeq2d 5260 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  F : B --> C ) )
 
Theoremfeq3d 5261 Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : X --> A  <->  F : X --> B ) )
 
Theoremfeq12d 5262 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 5263 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 5264 Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
 |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D ) 
 ->  ( F : A --> B 
 <->  G : C --> D ) )
 
Theoremfeq1i 5265 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F : A
 --> B  <->  G : A --> B )
 
Theoremfeq2i 5266 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
 |-  A  =  B   =>    |-  ( F : A
 --> C  <->  F : B --> C )
 
Theoremfeq23i 5267 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( F : A --> B 
 <->  F : C --> D )
 
Theoremfeq23d 5268 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( F : A --> B  <->  F : C --> D ) )
 
Theoremnff 5269 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 5270* 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 5271* 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 5272 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  F  Fn  A )
 
Theoremffnd 5273 A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  Fn  A )
 
Theoremdffn2 5274 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 5275 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Fun  F )
 
Theoremffund 5276 A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  Fun 
 F )
 
Theoremfrel 5277 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Rel  F )
 
Theoremfdm 5278 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  dom  F  =  A )
 
Theoremfdmd 5279 Deduction form of fdm 5278. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  dom 
 F  =  A )
 
Theoremfdmi 5280 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 |-  F : A --> B   =>    |-  dom  F  =  A
 
Theoremfrn 5281 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  ran  F  C_  B )
 
Theoremfrnd 5282 Deduction form of frn 5281. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ran 
 F  C_  B )
 
Theoremdffn3 5283 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 |-  ( F  Fn  A  <->  F : A --> ran  F )
 
Theoremfss 5284 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 5285 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 )
 
Theoremfssdmd 5286 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  D  C_  dom  F )   =>    |-  ( ph  ->  D  C_  A )
 
Theoremfssdm 5287 Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
 |-  D  C_  dom  F   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  D  C_  A )
 
Theoremfco 5288 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 5289 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 5290 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 5291 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 5292 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 5293 A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
 |-  ( F : A --> B  ->  ( F : dom  F --> B  /\  dom  F 
 C_  A ) )
 
Theoremopelf 5294 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 5295 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 5296 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 5297 Composition of two functions. (Contributed by NM, 22-May-2006.)
 |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B )
 
Theoremfssres 5298 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 )
 
Theoremfssresd 5299 Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  ( F  |`  C ) : C --> B )
 
Theoremfssres2 5300 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13341
  Copyright terms: Public domain < Previous  Next >