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Theorem List for Metamath Proof Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnsng 5301 Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. }  Fn  { A } )
 
Theoremfunsn 5302 A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 Fun  { <. A ,  B >. }
 
Theoremfunprg 5303 A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B ) 
 ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
 
Theoremfunpr 5304 A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  B  ->  Fun  { <. A ,  C >. ,  <. B ,  D >. } )
 
Theoremfuntp 5305 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) 
 ->  Fun  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. } )
 
Theoremfnsn 5306 Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 { <. A ,  B >. }  Fn  { A }
 
Theoremfnprg 5307 Domain of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )  /\  A  =/=  B ) 
 ->  { <. A ,  C >. ,  <. B ,  D >. }  Fn  { A ,  B } )
 
Theoremfntp 5308 A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) 
 ->  { <. A ,  D >. ,  <. B ,  E >. ,  <. C ,  F >. }  Fn  { A ,  B ,  C }
 )
 
Theoremfun0 5309 The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
 |- 
 Fun  (/)
 
Theoremfuncnvcnv 5310 The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
 |-  ( Fun  A  ->  Fun  `' `' A )
 
Theoremfuncnv2 5311* A simpler equivalence for single-rooted (see funcnv 5312). (Contributed by NM, 9-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y E* x  x A y )
 
Theoremfuncnv 5312* The converse of a class is a function iff the class is single-rooted, which means that for any  y in the range of  A there is at most one  x such that  x A
y. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5311 for a simpler version. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E* x  x A y )
 
Theoremfuncnv3 5313* A condition showing a class is single-rooted. (See funcnv 5312). (Contributed by NM, 26-May-2006.)
 |-  ( Fun  `' A  <->  A. y  e.  ran  A E! x  e.  dom  A  x A y )
 
Theoremfun2cnv 5314* 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  A is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
 |-  ( Fun  `' `' A 
 <-> 
 A. x E* y  x A y )
 
Theoremsvrelfun 5315 A single-valued relation is a function. (See fun2cnv 5314 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
 |-  ( Fun  A  <->  ( Rel  A  /\  Fun  `' `' A ) )
 
Theoremfncnv 5316* Single-rootedness (see funcnv 5312) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
 |-  ( `' ( R  i^i  ( A  X.  B ) )  Fn  B  <->  A. y  e.  B  E! x  e.  A  x R y )
 
Theoremfun11 5317* Two ways of stating that  A 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  `' `' A  /\  Fun  `' A )  <->  A. x A. y A. z A. w ( ( x A y 
 /\  z A w )  ->  ( x  =  z  <->  y  =  w ) ) )
 
Theoremfununi 5318* The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. f  e.  A  ( Fun  f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f )
 )  ->  Fun  U. A )
 
Theoremfuncnvuni 5319* The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5312 for "single-rooted" definition.) (Contributed by NM, 11-Aug-2004.)
 |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f
 ) )  ->  Fun  `' U. A )
 
Theoremfun11uni 5320* 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.)
 |-  ( A. f  e.  A  ( ( Fun  f  /\  Fun  `' f )  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f
 ) )  ->  ( Fun  U. A  /\  Fun  `'
 U. A ) )
 
Theoremfunin 5321 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 5322 The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
 |-  ( Fun  `' F  ->  Fun  `' ( F  |`  A ) )
 
Theoremfuncnvres 5323 The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
 |-  ( Fun  `' F  ->  `' ( F  |`  A )  =  ( `' F  |`  ( F " A ) ) )
 
Theoremcnvresid 5324 Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
 |-  `' (  _I  |`  A )  =  (  _I  |`  A )
 
Theoremfuncnvres2 5325 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 5326 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 5327 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 5328 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 )
 
Theoremimadif 5329 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 ) ) )
 
Theoremimain 5330 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 ) ) )
 
Theoremfunimaexg 5331 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 5332 The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4133. 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 5333* 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 5334* 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 5332. (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 5335 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
 
Theoremfneq2 5336 Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F  Fn  A  <->  F  Fn  B ) )
 
Theoremfneq1d 5337 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 5338 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 5339 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 ) )
 
Theoremfneq1i 5340 Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F  Fn  A 
 <->  G  Fn  A )
 
Theoremfneq2i 5341 Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
 |-  A  =  B   =>    |-  ( F  Fn  A 
 <->  F  Fn  B )
 
Theoremnffn 5342 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 5343 A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Fun  F )
 
Theoremfnrel 5344 A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  Fn  A  ->  Rel  F )
 
Theoremfndm 5345 The domain of a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F  Fn  A  ->  dom  F  =  A )
 
Theoremfunfni 5346 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 5347 A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
 |-  ( ( F  Fn  A  /\  F  Fn  B )  ->  A  =  B )
 
Theoremfnbr 5348 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 5349 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 5350* 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 5351* 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 5352 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 5353 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 5354 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 5355 A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
 |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
 
Theoremfnresdisj 5356 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 5357 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 5358 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 5359 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 5360 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 5361 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 5362* An equivalence for functionality of a restriction. Compare dffun8 5283. (Contributed by Mario Carneiro, 20-May-2015.)
 |-  ( ( F  |`  A )  Fn  A  <->  A. x  e.  A  E! y  x F y )
 
Theoremfnresi 5363 Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
 |-  (  _I  |`  A )  Fn  A
 
Theoremfnima 5364 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 5365 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 5366 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 5367 Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 27159. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( ( F  Fn  A  /\  B  C_  A )  ->  ( ( F
 " B )  =  (/) 
 <->  B  =  (/) ) )
 
Theoremdfmpt3 5368 Alternate definition for the "maps to" notation df-mpt 4081. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  ( x  e.  A  |->  B )  =  U_ x  e.  A  ( { x }  X.  { B }
 )
 
Theoremfnopabg 5369* 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 5370* 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 5371* 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 5372* 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 5373 A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( x  e.  (/)  |->  A )  =  (/)
 
Theoremfnmpti 5374* 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 5375* 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 5376 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 5377 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A --> B 
 <->  G : A --> B ) )
 
Theoremfeq2 5378 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A --> C 
 <->  F : B --> C ) )
 
Theoremfeq3 5379 Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C --> A 
 <->  F : C --> B ) )
 
Theoremfeq23 5380 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 5381 Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A --> B  <->  G : A --> B ) )
 
Theoremfeq2d 5382 Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A --> C  <->  F : B --> C ) )
 
Theoremfeq12d 5383 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 5384 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 ) )
 
Theoremfeq1i 5385 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  F  =  G   =>    |-  ( F : A
 --> B  <->  G : A --> B )
 
Theoremfeq2i 5386 Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
 |-  A  =  B   =>    |-  ( F : A
 --> C  <->  F : B --> C )
 
Theoremfeq23i 5387 Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  A  =  C   &    |-  B  =  D   =>    |-  ( F : A --> B 
 <->  F : C --> D )
 
Theoremfeq23d 5388 Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
 |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  ( F : A --> B  <->  F : C --> D ) )
 
Theoremnff 5389 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
 
Theoremelimf 5390 Eliminate a mapping hypothesis for the weak deduction theorem dedth 3608, when a special case  G : A --> B is provable, in order to convert  F : A --> B from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
 |-  G : A --> B   =>    |-  if ( F : A --> B ,  F ,  G ) : A --> B
 
Theoremffn 5391 A mapping is a function. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  F  Fn  A )
 
Theoremdffn2 5392 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 5393 A mapping is a function. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Fun  F )
 
Theoremfrel 5394 A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  Rel  F )
 
Theoremfdm 5395 The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
 |-  ( F : A --> B  ->  dom  F  =  A )
 
Theoremfdmi 5396 The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
 |-  F : A --> B   =>    |-  dom  F  =  A
 
Theoremfrn 5397 The range of a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A --> B  ->  ran  F  C_  B )
 
Theoremdffn3 5398 A function maps to its range. (Contributed by NM, 1-Sep-1999.)
 |-  ( F  Fn  A  <->  F : A --> ran  F )
 
Theoremfss 5399 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 )
 
Theoremfco 5400 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 )
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