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Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnsdomg 7001 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 15-Jun-1998.)
 |-  ( ( om  e.  _V 
 /\  A  e.  om )  ->  A  ~<  om )
 
Theoremisfiniteg 7002 A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( om  e.  _V  ->  ( A  e.  Fin  <->  A  ~<  om ) )
 
Theoreminfsdomnn 7003 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( om  ~<_  A  /\  B  e.  om )  ->  B  ~<  A )
 
Theoreminfn0 7004 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoremfin2inf 7005 This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless 
om exists. (Contributed by NM, 13-Nov-2003.)
 |-  ( A  ~<  om  ->  om  e.  _V )
 
Theoremunfilem1 7006* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |- 
 ran  F  =  (
 ( A  +o  B )  \  A )
 
Theoremunfilem2 7007* Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  A  e.  om   &    |-  B  e.  om   &    |-  F  =  ( x  e.  B  |->  ( A  +o  x ) )   =>    |-  F : B -1-1-onto-> ( ( A  +o  B )  \  A )
 
Theoremunfilem3 7008 Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 16-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  B  ~~  (
 ( A  +o  B )  \  A ) )
 
Theoremunfi 7009 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  u.  B )  e.  Fin )
 
Theoremunfir 7010 If a union is finite, the operands are finite. Converse of unfi 7009. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  u.  B )  e.  Fin  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremunfi2 7011 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7009 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7005). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
 
Theoremdifinf 7012 An infinite set  A minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( -.  A  e.  Fin  /\  B  e.  Fin )  ->  -.  ( A  \  B )  e. 
 Fin )
 
Theoremxpfi 7013 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  X.  B )  e.  Fin )
 
Theoremdomunfican 7014 A finite set union cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
 |-  ( ( ( A  e.  Fin  /\  B  ~~  A )  /\  ( ( A  i^i  X )  =  (/)  /\  ( B  i^i  Y )  =  (/) ) )  ->  ( ( A  u.  X )  ~<_  ( B  u.  Y ) 
 <->  X  ~<_  Y ) )
 
Theoreminfcntss 7015* Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.)
 |-  A  e.  _V   =>    |-  ( om  ~<_  A  ->  E. x ( x  C_  A  /\  x  ~~  om ) )
 
Theoremprfi 7016 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
 |- 
 { A ,  B }  e.  Fin
 
Theoremtpfi 7017 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
 |- 
 { A ,  B ,  C }  e.  Fin
 
Theoremfiint 7018* Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of 
A is in  A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally. (Contributed by NM, 22-Sep-2002.)
 |-  ( A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A  <->  A. x ( ( x 
 C_  A  /\  x  =/= 
 (/)  /\  x  e.  Fin )  ->  |^| x  e.  A ) )
 
Theoremfnfi 7019 A version of fnex 5593 for finite sets that does not require Replacement. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Fn  A  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremfodomfi 7020 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8033 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  ~<_  A )
 
Theoremfodomfib 7021* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8035 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)
 |-  ( A  e.  Fin  ->  ( ( A  =/=  (/)  /\  E. f  f : A -onto-> B )  <->  ( (/)  ~<  B  /\  B 
 ~<_  A ) ) )
 
Theoremfofinf1o 7022 Any surjection from one finite set to another of equal size must be a bijection. (Contributed by Mario Carneiro, 19-Aug-2014.)
 |-  ( ( F : A -onto-> B  /\  A  ~~  B  /\  B  e.  Fin )  ->  F : A -1-1-onto-> B )
 
Theoremfidomdm 7023 Any finite set dominates its domain. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
 |-  ( F  e.  Fin  ->  dom  F  ~<_  F )
 
Theoremdmfi 7024 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
 |-  ( A  e.  Fin  ->  dom  A  e.  Fin )
 
Theoremcnvfi 7025 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  `' A  e.  Fin )
 
Theoremrnfi 7026 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfofi 7027 If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.)
 |-  ( ( A  e.  Fin  /\  F : A -onto-> B )  ->  B  e.  Fin )
 
Theoremf1fi 7028 If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( B  e.  Fin  /\  F : A -1-1-> B )  ->  A  e.  Fin )
 
Theoremiunfi 7029* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7030. Note that  B depends on  x, i.e. can be thought of as  B ( x ). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin )  ->  U_ x  e.  A  B  e.  Fin )
 
Theoremunifi 7030 The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. (Contributed by NM, 22-Aug-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
 |-  ( ( A  e.  Fin  /\  A  C_  Fin )  ->  U. A  e.  Fin )
 
Theoremunifi2 7031* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7030 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7005). (Contributed by NM, 11-Mar-2006.)
 |-  ( ( A  ~<  om 
 /\  A. x  e.  A  x  ~<  om )  ->  U. A  ~<  om )
 
Theoremimafi 7032 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( Fun  F  /\  X  e.  Fin )  ->  ( F " X )  e.  Fin )
 
Theoremsuppfif1 7033 Formula building theorem for finite supports: rearranging the index set. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  ( ph  ->  ( `' F " ( _V  \  { Z } )
 )  e.  Fin )   &    |-  ( ph  ->  G : X -1-1-> Y )   =>    |-  ( ph  ->  ( `' ( F  o.  G ) " ( _V  \  { Z } ) )  e. 
 Fin )
 
Theorempwfilem 7034* Lemma for pwfi 7035. (Contributed by NM, 26-Mar-2007.)
 |-  F  =  ( c  e.  ~P b  |->  ( c  u.  { x } ) )   =>    |-  ( ~P b  e.  Fin  ->  ~P (
 b  u.  { x } )  e.  Fin )
 
Theorempwfi 7035 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.)
 |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
 
Theoremmapfi 7036 Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  e.  Fin  /\  B  e.  Fin )  ->  ( A  ^m  B )  e.  Fin )
 
Theoremixpfi 7037* A cross product of finitely many finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( A  e.  Fin  /\  A. x  e.  A  B  e.  Fin )  ->  X_ x  e.  A  B  e.  Fin )
 
Theoremixpfi2 7038* A cross product of finite sets such that all but finitely many are singletons is finite. (Note that  B ( x ) and 
D ( x ) are both possibly dependent on  x. ) (Contributed by Mario Carneiro, 25-Jan-2015.)
 |-  ( ph  ->  C  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  Fin )   &    |-  (
 ( ph  /\  x  e.  ( A  \  C ) )  ->  B  C_  { D } )   =>    |-  ( ph  ->  X_ x  e.  A  B  e.  Fin )
 
Theoremmptfi 7039* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  ->  ( x  e.  A  |->  B )  e.  Fin )
 
Theoremabrexfi 7040* An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014.)
 |-  ( A  e.  Fin  ->  { y  |  E. x  e.  A  y  =  B }  e.  Fin )
 
Theoremelfpw 7041 Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  ( A  e.  ( ~P B  i^i  Fin )  <->  ( A  C_  B  /\  A  e.  Fin ) )
 
Theoremunifpw 7042 A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |- 
 U. ( ~P A  i^i  Fin )  =  A
 
Theoremf1opwfi 7043* A one-to-one mapping induces a one-to-one mapping on finite subsets. (Contributed by Mario Carneiro, 25-Jan-2015.)
 |-  ( F : A -1-1-onto-> B  ->  ( b  e.  ( ~P A  i^i  Fin )  |->  ( F " b
 ) ) : ( ~P A  i^i  Fin )
 -1-1-onto-> ( ~P B  i^i  Fin ) )
 
Theoremfissuni 7044* A finite subset of a union is covered by finitely many elements. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( A  C_  U. B  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin ) A  C_  U. c
 )
 
Theoremfipreima 7045* Given a finite subset  A of the range of a function, there exists a finite subset of the domain whose image is  A. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( F  Fn  B  /\  A  C_  ran  F  /\  A  e.  Fin )  ->  E. c  e.  ( ~P B  i^i  Fin )
 ( F " c
 )  =  A )
 
Theoremfinsschain 7046* A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 17571 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)
 |-  ( ( ( A  =/=  (/)  /\ [ C.]  Or  A )  /\  ( B  e.  Fin  /\  B  C_  U. A ) )  ->  E. z  e.  A  B  C_  z
 )
 
Theoremindexfi 7047* If for every element of a finite indexing set  A there exists a corresponding element of another set  B, then there exists a finite subset of  B consisting only of those elements which are indexed by  A. Proven without the Axiom of Choice, unlike indexdom 25579. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
 |-  ( ( A  e.  Fin  /\  B  e.  M  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. c  e.  Fin  ( c  C_  B  /\  A. x  e.  A  E. y  e.  c  ph  /\ 
 A. y  e.  c  E. x  e.  A  ph ) )
 
2.4.33  Finite intersections
 
Syntaxcfi 7048 Extend class notation with the function whose value is the class of all the finite intersections of the elements of a given set.
 class  fi
 
Definitiondf-fi 7049* Function whose value is the class of all the finite intersections of the elements of  x. (Contributed by FL, 27-Apr-2008.)
 |- 
 fi  =  ( x  e.  _V  |->  { z  |  E. y  e.  ( ~P x  i^i  Fin )
 z  =  |^| y } )
 
Theoremfival 7050* The set of all the finite intersections of the elements of  A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( fi `  A )  =  { y  |  E. x  e.  ( ~P A  i^i  Fin )
 y  =  |^| x } )
 
Theoremelfi 7051* Specific properties of an element of 
( fi `  B
). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  ( ~P B  i^i  Fin ) A  =  |^| x ) )
 
Theoremelfi2 7052* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  ( B  e.  V  ->  ( A  e.  ( fi `  B )  <->  E. x  e.  (
 ( ~P B  i^i  Fin )  \  { (/) } ) A  =  |^| x ) )
 
Theoremelfir 7053 Sufficient condition for an element of  ( fi `  B ). (Contributed by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( B  e.  V  /\  ( A  C_  B  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^| A  e.  ( fi
 `  B ) )
 
Theoremintrnfi 7054 Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( ( B  e.  V  /\  ( F : A
 --> B  /\  A  =/=  (/)  /\  A  e.  Fin )
 )  ->  |^| ran  F  e.  ( fi `  B ) )
 
Theoremiinfi 7055* An indexed intersection of elements of  C is an element of the finite intersections of  C. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( ( C  e.  V  /\  ( A. x  e.  A  B  e.  C  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^|_
 x  e.  A  B  e.  ( fi `  C ) )
 
Theoremssfii 7056 Any element of a set  A is the intersection of a finite subset of  A. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( A  e.  V  ->  A  C_  ( fi `  A ) )
 
Theoremfi0 7057 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  (/) )  =  (/)
 
Theoremfieq0 7058 If  A is not empty, the class of all the finite intersections of  A is not empty either. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( A  =  (/)  <->  ( fi `  A )  =  (/) ) )
 
Theoremfiin 7059 The elements of  ( fi `  C ) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( A  e.  ( fi `  C ) 
 /\  B  e.  ( fi `  C ) ) 
 ->  ( A  i^i  B )  e.  ( fi `  C ) )
 
Theoremdffi2 7060* The set of finite intersections is the smallest set that contains  A and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( fi `  A )  =  |^| { z  |  ( A  C_  z  /\  A. x  e.  z  A. y  e.  z  ( x  i^i  y )  e.  z ) }
 )
 
Theoremfiss 7061 Subset relationship for function 
fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( ( B  e.  V  /\  A  C_  B )  ->  ( fi `  A )  C_  ( fi
 `  B ) )
 
Theoreminficl 7062* A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  ( A. x  e.  A  A. y  e.  A  ( x  i^i  y )  e.  A  <->  ( fi `  A )  =  A ) )
 
Theoremfipwuni 7063 The set of finite intersections of a set is contained in the powerset of the union of the elements of 
A. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
 |-  ( fi `  A )  C_  ~P U. A
 
Theoremfisn 7064 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( fi `  { A } )  =  { A }
 
Theoremfiuni 7065 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
 |-  ( A  e.  V  ->  U. A  =  U. ( fi `  A ) )
 
Theoremfipwss 7066 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( A  C_  ~P X  ->  ( fi `  A )  C_  ~P X )
 
Theoremelfiun 7067* A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)
 |-  ( ( B  e.  D  /\  C  e.  K )  ->  ( A  e.  ( fi `  ( B  u.  C ) )  <-> 
 ( A  e.  ( fi `  B )  \/  A  e.  ( fi
 `  C )  \/ 
 E. x  e.  ( fi `  B ) E. y  e.  ( fi `  C ) A  =  ( x  i^i  y ) ) ) )
 
Theoremdffi3 7068* The set of finite intersections can be "constructed" inductively by iterating binary intersection  om-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |-  R  =  ( u  e.  _V  |->  ran  (  y  e.  u ,  z  e.  u  |->  ( y  i^i  z ) ) )   =>    |-  ( A  e.  V  ->  ( fi `  A )  =  U. ( rec ( R ,  A ) " om ) )
 
Theoremfifo 7069* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
 |-  F  =  ( y  e.  ( ( ~P A  i^i  Fin )  \  { (/) } )  |->  |^| y )   =>    |-  ( A  e.  V  ->  F : ( ( ~P A  i^i  Fin )  \  { (/) } ) -onto->
 ( fi `  A ) )
 
2.4.34  Hall's marriage theorem
 
Theoremmarypha1lem 7070* Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)
 |-  ( A  e.  Fin  ->  ( b  e.  Fin  ->  A. c  e.  ~P  ( A  X.  b
 ) ( A. d  e.  ~P  A d  ~<_  ( c " d ) 
 ->  E. e  e.  ~P  c e : A -1-1-> _V ) ) )
 
Theoremmarypha1 7071* (Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pidgeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  C  C_  ( A  X.  B ) )   &    |-  ( ( ph  /\  d  C_  A )  ->  d  ~<_  ( C " d ) )   =>    |-  ( ph  ->  E. f  e.  ~P  C f : A -1-1-> B )
 
Theoremmarypha2lem1 7072* Lemma for marypha2 7076. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  ( F `  x ) )   =>    |-  T  C_  ( A  X.  U. ran  F )
 
Theoremmarypha2lem2 7073* Lemma for marypha2 7076. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  ( F `  x ) )   =>    |-  T  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) }
 
Theoremmarypha2lem3 7074* Lemma for marypha2 7076. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  ( F `  x ) )   =>    |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( G  C_  T  <->  A. x  e.  A  ( G `  x )  e.  ( F `  x ) ) )
 
Theoremmarypha2lem4 7075* Lemma for marypha2 7076. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  T  =  U_ x  e.  A  ( { x }  X.  ( F `  x ) )   =>    |-  ( ( F  Fn  A  /\  X  C_  A )  ->  ( T " X )  = 
 U. ( F " X ) )
 
Theoremmarypha2 7076* Version of marypha1 7071 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A --> Fin )   &    |-  (
 ( ph  /\  d  C_  A )  ->  d  ~<_  U. ( F " d ) )   =>    |-  ( ph  ->  E. g
 ( g : A -1-1-> _V 
 /\  A. x  e.  A  ( g `  x )  e.  ( F `  x ) ) )
 
2.4.35  Supremum
 
Syntaxcsup 7077 Extend class notation to include supremum of class  A. Here  R is ordinarily a relation that strictly orders class  B. For example,  R could be 'less than' and  B could be the set of real numbers.
 class  sup ( A ,  B ,  R )
 
Definitiondf-sup 7078* Define the supremum of class  A. It is meaningful when 
R is a relation that strictly orders  B and when the supremum exists. For example,  R could be 'less than',  B could be the set of real numbers, and  A could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrval 11599. See dfsup2 7079 for alternate definition not requiring dummy variables.

We will also use this notation for "infimum" by replacing  R with  `' R. (Contributed by NM, 22-May-1999.)

 |- 
 sup ( A ,  B ,  R )  =  U. { x  e.  B  |  ( A. y  e.  A  -.  x R y  /\  A. y  e.  B  (
 y R x  ->  E. z  e.  A  y R z ) ) }
 
Theoremdfsup2 7079 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
 |- 
 sup ( B ,  A ,  R )  =  U. ( A  \  ( ( `' R " B )  u.  ( R " ( A  \  ( `' R " B ) ) ) ) )
 
Theoremdfsup2OLD 7080 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 18-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 sup ( B ,  A ,  R )  =  U. ( A  \  ( ( `' R " B )  u.  (
 ( R  \  (
 ( `' R " B )  X.  _V )
 ) " A ) ) )
 
Theoremdfsup3OLD 7081 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 sup ( B ,  A ,  R )  =  U. ( A  \  ( ( `' R " B )  u.  ( R " ( A  \  ( `' R " B ) ) ) ) )
 
Theoremsupeq1 7082 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 7083 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 7084 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 
Theoremsupeq2 7085 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremnfsup 7086 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x R   =>    |-  F/_ x sup ( A ,  B ,  R )
 
Theoremsupmo 7087* Any class  B has at most one supremum in  A (where  R is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  E* x ( x  e.  A  /\  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) ) )
 
Theoremsupexd 7088 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoremsupeu 7089* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  E! x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupval2 7090* Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  ( iota_ x  e.  A ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) ) )
 
Theoremeqsup 7091* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  A. y  e.  B  -.  C R y  /\  A. y  e.  A  ( y R C  ->  E. z  e.  B  y R z ) ) 
 ->  sup ( B ,  A ,  R )  =  C ) )
 
Theoremeqsupd 7092* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  C R y )   &    |-  (
 ( ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. z  e.  B  y R z )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremsupcl 7093* A supremum belongs to its base class (closure law). (Contributed by NM, 12-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  A )
 
Theoremsupub 7094* A supremum is an upper bound. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  ( C  e.  B  ->  -. 
 sup ( B ,  A ,  R ) R C ) )
 
Theoremsuplub 7095* A supremum is the least upper bound. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  C R sup ( B ,  A ,  R ) )  ->  E. z  e.  B  C R z ) )
 
Theoremsuplub2 7096* Bidirectional form of suplub 7095. (Contributed by Mario Carneiro, 6-Sep-2014.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ( ph  /\  C  e.  A ) 
 ->  ( C R sup ( B ,  A ,  R )  <->  E. z  e.  B  C R z ) )
 
Theoremsupnub 7097* An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   =>    |-  ( ph  ->  (
 ( C  e.  A  /\  A. z  e.  B  -.  C R z ) 
 ->  -.  C R sup ( B ,  A ,  R ) ) )
 
Theoremsupex 7098 A supremum is a set. (Contributed by NM, 22-May-1999.)
 |-  R  Or  A   =>    |-  sup ( B ,  A ,  R )  e.  _V
 
Theoremsupmaxlem 7099* A set that contains a greatest element satisfies the antecedent in supremum theorems. This allows  sup ( A ,  B ,  R ) to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  ( ( C  e.  A  /\  C  e.  B  /\  A. z  e.  B  -.  C R z ) 
 ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupmax 7100* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  -.  C R y )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
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