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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunirnffid 7101 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 7102 Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( ( Fun  F  /\  X  e.  Fin )  ->  ( F " X )  e.  Fin )
 
Theoremsuppfif1 7103 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 7104* Lemma for pwfi 7105. (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 7105 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 7106 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 7107* 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 7108* 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 7109* A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.)
 |-  ( A  e.  Fin  ->  ( x  e.  A  |->  B )  e.  Fin )
 
Theoremabrexfi 7110* 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 7111 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 7112 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 7113* 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 7114* 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 7115* 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 7116* 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 17687 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 7117* 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 25766. (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 7118 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 7119* 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 7120* 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 7121* 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 7122* 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 7123 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 7124 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 7125* 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 7126 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 7127 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
 |-  ( fi `  (/) )  =  (/)
 
Theoremfieq0 7128 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 7129 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 7130* 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 7131 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 7132* 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 7133 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 7134 A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( fi `  { A } )  =  { A }
 
Theoremfiuni 7135 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 7136 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 7137* 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 7138* 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 7139* 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 7140* 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 7141* (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 7142* Lemma for marypha2 7146. 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 7143* Lemma for marypha2 7146. 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 7144* Lemma for marypha2 7146. 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 7145* Lemma for marypha2 7146. 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 7146* Version of marypha1 7141 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 7147 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 7148* 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 11673. See dfsup2 7149 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 7149 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 7150 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 7151 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 7152 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
 |-  ( B  =  C  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 )
 
Theoremsupeq1d 7153 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R ) )
 
Theoremsupeq1i 7154 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  B  =  C   =>    |-  sup ( B ,  A ,  R )  =  sup ( C ,  A ,  R )
 
Theoremsupeq2 7155 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( B  =  C  ->  sup ( A ,  B ,  R )  =  sup ( A ,  C ,  R )
 )
 
Theoremnfsup 7156 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 7157* 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  e.  A ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremsupexd 7158 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 7159* 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 7160* 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 7161* 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 7162* 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 7163* 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 7164* 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 7165* 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 7166* Bidirectional form of suplub 7165. (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 7167* 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 7168 A supremum is a set. (Contributed by NM, 22-May-1999.)
 |-  R  Or  A   =>    |-  sup ( B ,  A ,  R )  e.  _V
 
Theoremsupmaxlem 7169* 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 7170* 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 )
 
Theoremfisup2g 7171* A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
 |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  E. x  e.  B  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )
 
Theoremfisupcl 7172 A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
 |-  ( ( R  Or  A  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  A ) )  ->  sup ( B ,  A ,  R )  e.  B )
 
Theoremsuppr 7173 The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
 |-  ( ( R  Or  A  /\  B  e.  A  /\  C  e.  A ) 
 ->  sup ( { B ,  C } ,  A ,  R )  =  if ( C R B ,  B ,  C )
 )
 
Theoremsupsn 7174 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
 |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
 
Theoremsupisolem 7175* Lemma for supiso 7177. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   =>    |-  ( ( ph  /\  D  e.  A ) 
 ->  ( ( A. y  e.  C  -.  D R y  /\  A. y  e.  A  ( y R D  ->  E. z  e.  C  y R z ) )  <->  ( A. w  e.  ( F " C )  -.  ( F `  D ) S w 
 /\  A. w  e.  B  ( w S ( F `
  D )  ->  E. v  e.  ( F " C ) w S v ) ) ) )
 
Theoremsupisoex 7176* Lemma for supiso 7177. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  E. u  e.  B  ( A. w  e.  ( F " C )  -.  u S w 
 /\  A. w  e.  B  ( w S u  ->  E. v  e.  ( F " C ) w S v ) ) )
 
Theoremsupiso 7177* Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   &    |-  ( ph  ->  R  Or  A )   =>    |-  ( ph  ->  sup ( ( F " C ) ,  B ,  S )  =  ( F `  sup ( C ,  A ,  R ) ) )
 
2.4.36  Ordinal isomorphism, Hartog's theorem
 
Syntaxcoi 7178 Extend class definition to include the canonical order isomorphism to an ordinal.
 class OrdIso ( R ,  A )
 
Definitiondf-oi 7179* Define the canonical order isomorphism from the well-order  R on  A to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
 |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  (recs ( ( h  e. 
 _V  |->  ( iota_ v  e. 
 { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) )  |`  { x  e.  On  |  E. t  e.  A  A. z  e.  (recs ( ( h  e.  _V  |->  ( iota_ v  e.  { w  e.  A  |  A. j  e.  ran  h  j R w } A. u  e.  { w  e.  A  |  A. j  e.  ran  h  j R w }  -.  u R v ) ) ) " x ) z R t } ) ,  (/) )
 
Theoremdfoi 7180* Rewrite df-oi 7179 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  (
 iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  F  = recs ( G )   =>    |- OrdIso ( R ,  A )  =  if ( ( R  We  A  /\  R Se  A ) ,  ( F  |`  { x  e. 
 On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t } ) ,  (/) )
 
Theoremoieq1 7181 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( R  =  S  -> OrdIso ( R ,  A )  = OrdIso ( S ,  A ) )
 
Theoremoieq2 7182 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  ( A  =  B  -> OrdIso ( R ,  A )  = OrdIso ( R ,  B ) )
 
Theoremnfoi 7183 Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x R   &    |-  F/_ x A   =>    |-  F/_ xOrdIso ( R ,  A )
 
Theoremordiso2 7184 Generalize ordiso 7185 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 7185* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  =  B 
 <-> 
 E. f  f  Isom  _E 
 ,  _E  ( A ,  B ) ) )
 
Theoremordtypecbv 7186* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 26-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   =>    |- recs
 ( ( f  e. 
 _V  |->  ( iota_ s  e. 
 { y  e.  A  |  A. i  e.  ran  f  i R y } A. r  e.  { y  e.  A  |  A. i  e.  ran  f  i R y }  -.  r R s ) ) )  =  F
 
Theoremordtypelem1 7187* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  =  ( F  |`  T ) )
 
Theoremordtypelem2 7188* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  Ord  T )
 
Theoremordtypelem3 7189* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ph  /\  M  e.  ( T  i^i  dom  F ) )  ->  ( F `  M )  e. 
 { v  e.  { w  e.  A  |  A. j  e.  ( F " M ) j R w }  |  A. u  e.  { w  e.  A  |  A. j  e.  ( F " M ) j R w }  -.  u R v } )
 
Theoremordtypelem4 7190* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O : ( T  i^i  dom 
 F ) --> A )
 
Theoremordtypelem5 7191* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  ( Ord  dom  O  /\  O : dom  O --> A ) )
 
Theoremordtypelem6 7192* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ph  /\  M  e.  dom  O )  ->  ( N  e.  M  ->  ( O `  N ) R ( O `  M ) ) )
 
Theoremordtypelem7 7193* Lemma for ordtype 7201. 
ran  O is an initial segment of  A under the well-order  R. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ( ( ph  /\  N  e.  A ) 
 /\  M  e.  dom  O )  ->  ( ( O `  M ) R N  \/  N  e.  ran 
 O ) )
 
Theoremordtypelem8 7194* Lemma for ordtype 7201. (Contributed by Mario Carneiro, 17-Oct-2009.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
 
Theoremordtypelem9 7195* Lemma for ordtype 7201. Either the function OrdIso is an isomorphism onto all of  A, or OrdIso is not a set, which by oif 7199 implies that either  ran  O 
C_  A is a proper class or  dom  O  =  On. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   &    |-  ( ph  ->  O  e.  _V )   =>    |-  ( ph  ->  O 
 Isom  _E  ,  R  ( dom  O ,  A ) )
 
Theoremordtypelem10 7196* Lemma for ordtype 7201. Using ax-rep 4091, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = recs ( G )   &    |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }   &    |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C A. u  e.  C  -.  u R v ) )   &    |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }   &    |-  O  = OrdIso ( R ,  A )   &    |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  R Se  A )   =>    |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
 
Theoremoi0 7197 Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( -.  ( R  We  A  /\  R Se  A )  ->  F  =  (/) )
 
Theoremoicl 7198 The order type of the well-order  R on  A is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  Ord  dom  F
 
Theoremoif 7199 The order isomorphism of the well-order  R on  A is a function. (Contributed by Mario Carneiro, 23-May-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  F : dom  F --> A
 
Theoremoiiso2 7200 The order isomorphism of the well-order  R on  A is an isomorphism onto  ran  O (which is a subset of  A by oif 7199). (Contributed by Mario Carneiro, 25-Jun-2015.)
 |-  F  = OrdIso ( R ,  A )   =>    |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E 
 ,  R  ( dom 
 F ,  ran  F ) )
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