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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuplubti 7101* A supremum is the least upper bound. See also supclti 7099 and supubti 7100. (Contributed by Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( 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 ) )
 
Theoremsuplub2ti 7102* Bidirectional form of suplubti 7101. (Contributed by Jim Kingdon, 17-Jan-2022.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( 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  ->  R  Or  A )   &    |-  ( ph  ->  B  C_  A )   =>    |-  ( ( ph  /\  C  e.  A )  ->  ( C R sup ( B ,  A ,  R ) 
 <-> 
 E. z  e.  B  C R z ) )
 
Theoremsupelti 7103* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  C  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  C  C_  A )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  e.  C )
 
Theoremsup00 7104 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- 
 sup ( B ,  (/)
 ,  R )  =  (/)
 
Theoremsupmaxti 7105* 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 Jim Kingdon, 24-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  C R y )   =>    |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
 
Theoremsupsnti 7106* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  ->  sup ( { B } ,  A ,  R )  =  B )
 
Theoremisotilem 7107* Lemma for isoti 7108. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( F  Isom  R ,  S  ( A ,  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  =  y  <->  ( -.  x S y  /\  -.  y S x ) )  ->  A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) ) )
 
Theoremisoti 7108* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( F  Isom  R ,  S  ( A ,  B )  ->  ( A. u  e.  A  A. v  e.  A  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) )  <->  A. u  e.  B  A. v  e.  B  ( u  =  v  <->  ( -.  u S v  /\  -.  v S u ) ) ) )
 
Theoremsupisolem 7109* Lemma for supisoti 7111. (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 7110* Lemma for supisoti 7111. (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 ) ) )
 
Theoremsupisoti 7111* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
 |-  ( 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  /\  ( u  e.  A  /\  v  e.  A ) )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  sup ( ( F " C ) ,  B ,  S )  =  ( F ` 
 sup ( C ,  A ,  R )
 ) )
 
Theoreminfeq1 7112 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1d 7113 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  = inf ( C ,  A ,  R ) )
 
Theoreminfeq1i 7114 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  B  =  C   =>    |- inf ( B ,  A ,  R )  = inf ( C ,  A ,  R )
 
Theoreminfeq2 7115 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( B  =  C  -> inf ( A ,  B ,  R )  = inf ( A ,  C ,  R ) )
 
Theoreminfeq3 7116 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( R  =  S  -> inf ( A ,  B ,  R )  = inf ( A ,  B ,  S ) )
 
Theoreminfeq123d 7117 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  ( ph  ->  A  =  D )   &    |-  ( ph  ->  B  =  E )   &    |-  ( ph  ->  C  =  F )   =>    |-  ( ph  -> inf ( A ,  B ,  C )  = inf ( D ,  E ,  F ) )
 
Theoremnfinf 7118 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x R   =>    |-  F/_ xinf ( A ,  B ,  R )
 
Theoremcnvinfex 7119* Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( 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 ) ) )
 
Theoremcnvti 7120* If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u `' R v  /\  -.  v `' R u ) ) )
 
Theoremeqinfti 7121* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  A. y  e.  B  -.  y R C  /\  A. y  e.  A  ( C R y  ->  E. z  e.  B  z R y ) )  -> inf ( B ,  A ,  R )  =  C )
 )
 
Theoremeqinftid 7122* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ( ph  /\  y  e.  B ) 
 ->  -.  y R C )   &    |-  ( ( ph  /\  (
 y  e.  A  /\  C R y ) ) 
 ->  E. z  e.  B  z R y )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
 
Theoreminfvalti 7123* Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  ( iota_ x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) ) )
 
Theoreminfclti 7124* An infimum belongs to its base class (closure law). See also inflbti 7125 and infglbti 7126. (Contributed by Jim Kingdon, 17-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  e.  A )
 
Theoreminflbti 7125* An infimum is a lower bound. See also infclti 7124 and infglbti 7126. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( C  e.  B  ->  -.  C Rinf ( B ,  A ,  R ) ) )
 
Theoreminfglbti 7126* An infimum is the greatest lower bound. See also infclti 7124 and inflbti 7125. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\ inf ( B ,  A ,  R ) R C )  ->  E. z  e.  B  z R C ) )
 
Theoreminfnlbti 7127* A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  ( ( C  e.  A  /\  A. z  e.  B  -.  z R C ) 
 ->  -. inf ( B ,  A ,  R ) R C ) )
 
Theoreminfminti 7128* The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  C  e.  B )   &    |-  (
 ( ph  /\  y  e.  B )  ->  -.  y R C )   =>    |-  ( ph  -> inf ( B ,  A ,  R )  =  C )
 
Theoreminfmoti 7129* Any class  B has at most one infimum in  A (where  R is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  ->  E* x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
 
Theoreminfeuti 7130* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )   =>    |-  ( ph  ->  E! x  e.  A  (
 A. y  e.  B  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  B  z R y ) ) )
 
Theoreminfsnti 7131* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A )
 )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   &    |-  ( ph  ->  B  e.  A )   =>    |-  ( ph  -> inf ( { B } ,  A ,  R )  =  B )
 
Theoreminf00 7132 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
 |- inf
 ( B ,  (/) ,  R )  =  (/)
 
Theoreminfisoti 7133* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
 |-  ( ph  ->  F  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  C  -.  y R x  /\  A. y  e.  A  ( x R y  ->  E. z  e.  C  z R y ) ) )   &    |-  ( ( ph  /\  ( u  e.  A  /\  v  e.  A ) )  ->  ( u  =  v  <->  ( -.  u R v  /\  -.  v R u ) ) )   =>    |-  ( ph  -> inf ( ( F " C ) ,  B ,  S )  =  ( F ` inf ( C ,  A ,  R ) ) )
 
Theoremsupex2g 7134 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
 
Theoreminfex2g 7135 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
 |-  ( A  e.  C  -> inf ( B ,  A ,  R )  e.  _V )
 
2.6.35  Ordinal isomorphism
 
Theoremordiso2 7136 Generalize ordiso 7137 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
 |-  ( ( F  Isom  _E 
 ,  _E  ( A ,  B )  /\  Ord 
 A  /\  Ord  B ) 
 ->  A  =  B )
 
Theoremordiso 7137* 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 ) ) )
 
2.6.36  Disjoint union
 
2.6.36.1  Disjoint union
 
Syntaxcdju 7138 Extend class notation to include disjoint union of two classes.
 class  ( A B )
 
Definitiondf-dju 7139 Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of  A or  B, tagging each one with whether it came from  A or  B. (Contributed by Jim Kingdon, 20-Jun-2022.)
 |-  ( A B )  =  ( ( { (/) }  X.  A )  u.  ( { 1o }  X.  B ) )
 
Theoremdjueq12 7140 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A C )  =  ( B D ) )
 
Theoremdjueq1 7141 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( A C )  =  ( B C )
 )
 
Theoremdjueq2 7142 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  ( A  =  B  ->  ( C A )  =  ( C B )
 )
 
Theoremnfdju 7143 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A B )
 
Theoremdjuex 7144 The disjoint union of sets is a set. See also the more precise djuss 7171. (Contributed by AV, 28-Jun-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  e.  _V )
 
Theoremdjuexb 7145 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V ) 
 <->  ( A B )  e.  _V )
 
2.6.36.2  Left and right injections of a disjoint union

In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form  (inl  |`  A ) and  (inr  |`  B ).

 
Syntaxcinl 7146 Extend class notation to include left injection of a disjoint union.
 class inl
 
Syntaxcinr 7147 Extend class notation to include right injection of a disjoint union.
 class inr
 
Definitiondf-inl 7148 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inl 
 =  ( x  e. 
 _V  |->  <. (/) ,  x >. )
 
Definitiondf-inr 7149 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inr 
 =  ( x  e. 
 _V  |->  <. 1o ,  x >. )
 
Theoremdjulclr 7150 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
 |-  ( C  e.  A  ->  ( (inl  |`  A ) `
  C )  e.  ( A B )
 )
 
Theoremdjurclr 7151 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
 |-  ( C  e.  B  ->  ( (inr  |`  B ) `
  C )  e.  ( A B )
 )
 
Theoremdjulcl 7152 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B )
 )
 
Theoremdjurcl 7153 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  B  ->  (inr `  C )  e.  ( A B )
 )
 
Theoremdjuf1olem 7154* Lemma for djulf1o 7159 and djurf1o 7160. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
 |-  X  e.  _V   &    |-  F  =  ( x  e.  A  |->  <. X ,  x >. )   =>    |-  F : A -1-1-onto-> ( { X }  X.  A )
 
Theoremdjuf1olemr 7155* Lemma for djulf1or 7157 and djurf1or 7158. For a version of this lemma with  F defined on  A and no restriction in the conclusion, see djuf1olem 7154. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
 |-  X  e.  _V   &    |-  F  =  ( x  e.  _V  |->  <. X ,  x >. )   =>    |-  ( F  |`  A ) : A -1-1-onto-> ( { X }  X.  A )
 
Theoremdjulclb 7156 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( C  e.  V  ->  ( C  e.  A  <->  (inl `  C )  e.  ( A B ) ) )
 
Theoremdjulf1or 7157 The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
 |-  (inl  |`  A ) : A -1-1-onto-> ( { (/) }  X.  A )
 
Theoremdjurf1or 7158 The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
 |-  (inr  |`  A ) : A -1-1-onto-> ( { 1o }  X.  A )
 
Theoremdjulf1o 7159 The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
 |- inl : _V
 -1-1-onto-> ( { (/) }  X.  _V )
 
Theoremdjurf1o 7160 The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
 |- inr : _V
 -1-1-onto-> ( { 1o }  X.  _V )
 
Theoreminresflem 7161* Lemma for inlresf1 7162 and inrresf1 7163. (Contributed by BJ, 4-Jul-2022.)
 |-  F : A -1-1-onto-> ( { X }  X.  A )   &    |-  ( x  e.  A  ->  ( F `  x )  e.  B )   =>    |-  F : A -1-1-> B
 
Theoreminlresf1 7162 The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
 |-  (inl  |`  A ) : A -1-1-> ( A B )
 
Theoreminrresf1 7163 The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
 |-  (inr  |`  B ) : B -1-1-> ( A B )
 
Theoremdjuinr 7164 The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 7194 and djufun 7205) while the simpler statement  |-  ( ran inl  i^i 
ran inr )  =  (/) is easily recovered from it by substituting  _V for both  A and  B as done in casefun 7186). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
 |-  ( ran  (inl  |`  A )  i^i  ran  (inr  |`  B ) )  =  (/)
 
Theoremdjuin 7165 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
 |-  ( (inl " A )  i^i  (inr " B ) )  =  (/)
 
Theoreminl11 7166 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (inl `  A )  =  (inl `  B )  <->  A  =  B ) )
 
Theoremdjuunr 7167 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
 |-  ( ran  (inl  |`  A )  u.  ran  (inr  |`  B ) )  =  ( A B )
 
Theoremdjuun 7168 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
 |-  ( (inl " A )  u.  (inr " B ) )  =  ( A B )
 
Theoremeldju 7169* Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
 |-  ( C  e.  ( A B )  <->  ( E. x  e.  A  C  =  ( (inl  |`  A ) `  x )  \/  E. x  e.  B  C  =  ( (inr  |`  B ) `  x ) ) )
 
Theoremdjur 7170* A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
 |-  ( C  e.  ( A B )  <->  ( E. x  e.  A  C  =  (inl `  x )  \/  E. x  e.  B  C  =  (inr `  x )
 ) )
 
2.6.36.3  Universal property of the disjoint union
 
Theoremdjuss 7171 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
 |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )
 
Theoremeldju1st 7172 The first component of an element of a disjoint union is either  (/) or  1o. (Contributed by AV, 26-Jun-2022.)
 |-  ( X  e.  ( A B )  ->  (
 ( 1st `  X )  =  (/)  \/  ( 1st `  X )  =  1o ) )
 
Theoremeldju2ndl 7173 The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
 |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =  (/) )  ->  ( 2nd `  X )  e.  A )
 
Theoremeldju2ndr 7174 The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
 |-  ( ( X  e.  ( A B )  /\  ( 1st `  X )  =/= 
 (/) )  ->  ( 2nd `  X )  e.  B )
 
Theorem1stinl 7175 The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
 |-  ( X  e.  V  ->  ( 1st `  (inl `  X ) )  =  (/) )
 
Theorem2ndinl 7176 The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
 |-  ( X  e.  V  ->  ( 2nd `  (inl `  X ) )  =  X )
 
Theorem1stinr 7177 The first component of the value of a right injection is  1o. (Contributed by AV, 27-Jun-2022.)
 |-  ( X  e.  V  ->  ( 1st `  (inr `  X ) )  =  1o )
 
Theorem2ndinr 7178 The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
 |-  ( X  e.  V  ->  ( 2nd `  (inr `  X ) )  =  X )
 
Theoremdjune 7179 Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl `  A )  =/=  (inr `  B ) )
 
Theoremupdjudhf 7180* The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
 |-  ( ph  ->  F : A --> C )   &    |-  ( ph  ->  G : B --> C )   &    |-  H  =  ( x  e.  ( A B )  |->  if (
 ( 1st `  x )  =  (/) ,  ( F `
  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )   =>    |-  ( ph  ->  H :
 ( A B ) --> C )
 
Theoremupdjudhcoinlf 7181* The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
 |-  ( ph  ->  F : A --> C )   &    |-  ( ph  ->  G : B --> C )   &    |-  H  =  ( x  e.  ( A B )  |->  if (
 ( 1st `  x )  =  (/) ,  ( F `
  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )   =>    |-  ( ph  ->  ( H  o.  (inl  |`  A ) )  =  F )
 
Theoremupdjudhcoinrg 7182* The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
 |-  ( ph  ->  F : A --> C )   &    |-  ( ph  ->  G : B --> C )   &    |-  H  =  ( x  e.  ( A B )  |->  if (
 ( 1st `  x )  =  (/) ,  ( F `
  ( 2nd `  x ) ) ,  ( G `  ( 2nd `  x ) ) ) )   =>    |-  ( ph  ->  ( H  o.  (inr  |`  B ) )  =  G )
 
Theoremupdjud 7183* Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
 |-  ( ph  ->  F : A --> C )   &    |-  ( ph  ->  G : B --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   =>    |-  ( ph  ->  E! h ( h :
 ( A B ) --> C  /\  ( h  o.  (inl  |`  A ) )  =  F  /\  ( h  o.  (inr  |`  B ) )  =  G ) )
 
Syntaxcdjucase 7184 Syntax for the "case" construction.
 class case ( R ,  S )
 
Definitiondf-case 7185 The "case" construction: if  F : A --> X and  G : B --> X are functions, then case ( F ,  G
) : ( A B ) --> X is the natural function obtained by a definition by cases, hence the name. It is the unique function whose existence is asserted by the universal property of disjoint unions updjud 7183. The definition is adapted to make sense also for binary relations (where the universal property also holds). (Contributed by MC and BJ, 10-Jul-2022.)
 |- case
 ( R ,  S )  =  ( ( R  o.  `'inl )  u.  ( S  o.  `'inr ) )
 
Theoremcasefun 7186 The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  Fun  F )   &    |-  ( ph  ->  Fun  G )   =>    |-  ( ph  ->  Fun case ( F ,  G ) )
 
Theoremcasedm 7187 The domain of the "case" construction is the disjoint union of the domains. TODO (although less important):  |-  ran case ( F ,  G )  =  ( ran  F  u.  ran  G ). (Contributed by BJ, 10-Jul-2022.)
 |- 
 dom case ( F ,  G )  =  ( dom  F dom  G )
 
Theoremcaserel 7188 The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)
 |- case
 ( R ,  S )  C_  ( ( dom 
 R dom  S )  X.  ( ran  R  u.  ran  S ) )
 
Theoremcasef 7189 The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  F : A --> X )   &    |-  ( ph  ->  G : B --> X )   =>    |-  ( ph  -> case ( F ,  G ) : ( A B ) --> X )
 
Theoremcaseinj 7190 The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  Fun  `' R )   &    |-  ( ph  ->  Fun  `' S )   &    |-  ( ph  ->  ( ran  R  i^i  ran  S )  =  (/) )   =>    |-  ( ph  ->  Fun  `'case ( R ,  S ) )
 
Theoremcasef1 7191 The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
 |-  ( ph  ->  F : A -1-1-> X )   &    |-  ( ph  ->  G : B -1-1-> X )   &    |-  ( ph  ->  ( ran  F  i^i  ran  G )  =  (/) )   =>    |-  ( ph  -> case ( F ,  G ) : ( A B ) -1-1-> X )
 
Theoremcaseinl 7192 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
 |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  Fun 
 G )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (case ( F ,  G ) `  (inl `  A ) )  =  ( F `  A ) )
 
Theoremcaseinr 7193 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
 |-  ( ph  ->  Fun  F )   &    |-  ( ph  ->  G  Fn  B )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  (case ( F ,  G ) `  (inr `  A ) )  =  ( G `  A ) )
 
2.6.36.4  Dominance and equinumerosity properties of disjoint union
 
Theoremdjudom 7194 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
 |-  ( ( A  ~<_  B  /\  C 
 ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
 
Theoremomp1eomlem 7195* Lemma for omp1eom 7196. (Contributed by Jim Kingdon, 11-Jul-2023.)
 |-  F  =  ( x  e.  om  |->  if ( x  =  (/) ,  (inr `  x ) ,  (inl ` 
 U. x ) ) )   &    |-  S  =  ( x  e.  om  |->  suc 
 x )   &    |-  G  = case ( S ,  (  _I  |` 
 1o ) )   =>    |-  F : om -1-1-onto-> ( om 1o )
 
Theoremomp1eom 7196 Adding one to  om. (Contributed by Jim Kingdon, 10-Jul-2023.)
 |-  ( om 1o )  ~~  om
 
Theoremendjusym 7197 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B ) 
 ~~  ( B A ) )
 
Theoremeninl 7198 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inl " A )  ~~  A )
 
Theoremeninr 7199 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inr " A )  ~~  A )
 
Theoremdifinfsnlem 7200* Lemma for difinfsn 7201. The case where we need to swap  B and  (inr `  (/) ) in building the mapping  G. (Contributed by Jim Kingdon, 9-Aug-2023.)
 |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  F : ( om 1o ) -1-1-> A )   &    |-  ( ph  ->  ( F `  (inr `  (/) ) )  =/=  B )   &    |-  G  =  ( n  e.  om  |->  if (
 ( F `  (inl `  n ) )  =  B ,  ( F `
  (inr `  (/) ) ) ,  ( F `  (inl `  n ) ) ) )   =>    |-  ( ph  ->  G : om -1-1-> ( A  \  { B } ) )
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