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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac6sfi 7101* A version of ac6s 8111 for finite sets. (Contributed by Jeffrey Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( y  =  ( f `  x ) 
 ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  Fin  /\  A. x  e.  A  E. y  e.  B  ph )  ->  E. f ( f : A --> B  /\  A. x  e.  A  ps ) )
 
Theoremfrfi 7102 A partial order is well-founded on a finite set. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Po  A  /\  A  e.  Fin )  ->  R  Fr  A )
 
Theoremfimax2g 7103* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
 
Theoremfimaxg 7104* A finite set has a maximum under a total order. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  A. y  e.  A  ( x  =/=  y  ->  y R x ) )
 
Theoremfisupg 7105* Lemma showing existence and closure of supremum of a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( R  Or  A  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A. y  e.  A  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  A  y R z ) ) )
 
Theoremwofi 7106 A total order on a finite set is a well order. (Contributed by Jeff Madsen, 18-Jun-2010.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( R  Or  A  /\  A  e.  Fin )  ->  R  We  A )
 
Theoremordunifi 7107 The maximum of a finite collection of ordinals is in the set. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)
 |-  ( ( A  C_  On  /\  A  e.  Fin  /\  A  =/=  (/) )  ->  U. A  e.  A )
 
Theoremnnunifi 7108 The union (supremum) of a finite set of finite ordinals is a finite ordinal. (Contributed by Stefan O'Rear, 5-Nov-2014.)
 |-  ( ( S  C_  om 
 /\  S  e.  Fin )  ->  U. S  e.  om )
 
Theoremunblem1 7109* Lemma for unbnn 7113. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( ( B 
 C_  om  /\  A. x  e.  om  E. y  e.  B  x  e.  y
 )  /\  A  e.  B )  ->  |^| ( B  \  suc  A )  e.  B )
 
Theoremunblem2 7110* Lemma for unbnn 7113. The value of the function  F belongs to the unbounded set of natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  A ) )
 
Theoremunblem3 7111* Lemma for unbnn 7113. The value of the function  F is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  (
 z  e.  om  ->  ( F `  z )  e.  ( F `  suc  z ) ) )
 
Theoremunblem4 7112* Lemma for unbnn 7113. The function  F maps the set of natural numbers one-to-one to the set of unbounded natural numbers  A. (Contributed by NM, 3-Dec-2003.)
 |-  F  =  ( rec ( ( x  e. 
 _V  |->  |^| ( A  \  suc  x ) ) , 
 |^| A )  |`  om )   =>    |-  ( ( A  C_  om 
 /\  A. w  e.  om  E. v  e.  A  w  e.  v )  ->  F : om -1-1-> A )
 
Theoremunbnn 7113* Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 7359 for a stronger version without the first assumption. (Contributed by NM, 3-Dec-2003.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  e.  y )  ->  A  ~~ 
 om )
 
Theoremunbnn2 7114* Version of unbnn 7113 that does not require a strict upper bound. (Contributed by NM, 24-Apr-2004.)
 |-  ( ( om  e.  _V 
 /\  A  C_  om  /\  A. x  e.  om  E. y  e.  A  x  C_  y )  ->  A  ~~ 
 om )
 
Theoremisfinite2 7115 Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity. (Contributed by NM, 24-Apr-2004.)
 |-  ( A  ~<  om  ->  A  e.  Fin )
 
Theoremnnsdomg 7116 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 7117 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 7118 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 7119 An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
 |-  ( om  ~<_  A  ->  A  =/=  (/) )
 
Theoremfin2inf 7120 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 7121* 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 7122* 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 7123 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 7124 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 7125 If a union is finite, the operands are finite. Converse of unfi 7124. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  u.  B )  e.  Fin  ->  ( A  e.  Fin  /\  B  e.  Fin )
 )
 
Theoremunfi2 7126 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 7124 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7120). (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 27-Apr-2015.)
 |-  ( ( A  ~<  om 
 /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
 
Theoremdifinf 7127 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 7128 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 7129 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 7130* 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 7131 An unordered pair is finite. (Contributed by NM, 22-Aug-2008.)
 |- 
 { A ,  B }  e.  Fin
 
Theoremtpfi 7132 An unordered triple is finite. (Contributed by Mario Carneiro, 28-Sep-2013.)
 |- 
 { A ,  B ,  C }  e.  Fin
 
Theoremfiint 7133* 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 7134 A version of fnex 5741 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 7135 An onto function implies dominance of domain over range, for finite sets. Unlike fodom 8149 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 7136* Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 8151 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 7137 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 7138 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 7139 The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.)
 |-  ( A  e.  Fin  ->  dom  A  e.  Fin )
 
Theoremcnvfi 7140 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  `' A  e.  Fin )
 
Theoremrnfi 7141 The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  ( A  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfofi 7142 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 7143 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 7144* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 7145. 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 7145 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 7146* The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 7145 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 7120). (Contributed by NM, 11-Mar-2006.)
 |-  ( ( A  ~<  om 
 /\  A. x  e.  A  x  ~<  om )  ->  U. A  ~<  om )
 
Theoremunirnffid 7147 The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  F : T --> Fin )   &    |-  ( ph  ->  T  e.  Fin )   =>    |-  ( ph  ->  U.
 ran  F  e.  Fin )
 
Theoremimafi 7148 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( Fun  F  /\  X  e.  Fin )  ->  ( F " X )  e.  Fin )
 
Theoremsuppfif1 7149 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 7150* Lemma for pwfi 7151. (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 7151 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 7152 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 7153* 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 7154* 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 7155* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  ->  ( x  e.  A  |->  B )  e.  Fin )
 
Theoremabrexfi 7156* 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 7157 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 7158 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 7159* 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 7160* 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 7161* 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 7162* 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 17739 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 7163* 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 26413. (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 ) )
 
2.4.35  Finite intersections
 
Syntaxcfi 7164 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 7165* 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 7166* 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 7167* 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 7168* 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 7169 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 7170 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 7171* 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 7172 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 7173 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  (/) )  =  (/)
 
Theoremfieq0 7174 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 7175 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 7176* 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 7177 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 7178* 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 7179 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 7180 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( fi `  { A } )  =  { A }
 
Theoremfiuni 7181 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 7182 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 7183* 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 7184* 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 7185* 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 ) )
 
2.4.36  Hall's marriage theorem
 
Theoremmarypha1lem 7186* 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 7187* (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 7188* Lemma for marypha2 7192. 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 7189* Lemma for marypha2 7192. 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 7190* Lemma for marypha2 7192. 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 7191* Lemma for marypha2 7192. 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 7192* Version of marypha1 7187 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 ) ) )
 
2.4.37  Supremum
 
Syntaxcsup 7193 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 7194* 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 11722. See dfsup2 7195 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 7195 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 7196 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 7197 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 7198 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 7199 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 7200 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
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