P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eqsuptid 7301*	Sufficient condition for an element to be equal to the supremum. (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 /\ 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 )
Theorem	supclti 7302*	A supremum belongs to its base class (closure law). See also supubti 7303 and suplubti 7304. (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 -> sup ( B , A , R ) e. A )
Theorem	supubti 7303*	A supremum is an upper bound. See also supclti 7302 and suplubti 7304. This proof demonstrates how to expand an iota-based definition (df-iota 5317) using riotacl2 6026. (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. B -> -. sup ( B , A , R ) R C ) )
Theorem	suplubti 7304*	A supremum is the least upper bound. See also supclti 7302 and supubti 7303. (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 ) )
Theorem	suplub2ti 7305*	Bidirectional form of suplubti 7304. (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 ) )
Theorem	supelti 7306*	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 )
Theorem	sup00 7307	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- sup ( B , (/) , R ) = (/)
Theorem	supmaxti 7308*	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 )
Theorem	supsnti 7309*	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 )
Theorem	isotilem 7310*	Lemma for isoti 7311. (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 ) ) ) )
Theorem	isoti 7311*	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 ) ) ) )
Theorem	supisolem 7312*	Lemma for supisoti 7314. (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 ) ) ) )
Theorem	supisoex 7313*	Lemma for supisoti 7314. (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 ) ) )
Theorem	supisoti 7314*	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 ) ) )
Theorem	infeq1 7315	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1d 7316	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> B = C )   =>    |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1i 7317	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
|- B = C   =>    |- inf ( B , A , R ) = inf ( C , A , R )
Theorem	infeq2 7318	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )
Theorem	infeq3 7319	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )
Theorem	infeq123d 7320	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 ) )
Theorem	nfinf 7321	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ xinf ( A , B , R )
Theorem	cnvinfex 7322*	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 ) ) )
Theorem	cnvti 7323*	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 ) ) )
Theorem	eqinfti 7324*	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 ) )
Theorem	eqinftid 7325*	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 )
Theorem	infvalti 7326*	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 ) ) ) )
Theorem	infclti 7327*	An infimum belongs to its base class (closure law). See also inflbti 7328 and infglbti 7329. (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 )
Theorem	inflbti 7328*	An infimum is a lower bound. See also infclti 7327 and infglbti 7329. (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 ) ) )
Theorem	infglbti 7329*	An infimum is the greatest lower bound. See also infclti 7327 and inflbti 7328. (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 ) )
Theorem	infnlbti 7330*	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 ) )
Theorem	infminti 7331*	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 )
Theorem	infmoti 7332*	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 ) ) )
Theorem	infeuti 7333*	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 ) ) )
Theorem	infsnti 7334*	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 )
Theorem	inf00 7335	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 7336*	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 ) ) )
Theorem	supex2g 7337	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. C -> sup ( B , A , R ) e. _V )
Theorem	infex2g 7338	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )
2.6.38  Ordinal isomorphism
Theorem	ordiso2 7339	Generalize ordiso 7340 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 7340*	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 ) ) )
2.6.39  Disjoint union
2.6.39.1  Disjoint union
Syntax	cdju 7341	Extend class notation to include disjoint union of two classes.
class ( A ⊔ B )
Definition	df-dju 7342	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 ) )
Theorem	djueq12 7343	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )
Theorem	djueq1 7344	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( A ⊔ C ) = ( B ⊔ C ) )
Theorem	djueq2 7345	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( C ⊔ A ) = ( C ⊔ B ) )
Theorem	nfdju 7346	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A ⊔ B )
Theorem	djuex 7347	The disjoint union of sets is a set. See also the more precise djuss 7374. (Contributed by AV, 28-Jun-2022.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) e. _V )
Theorem	djuexb 7348	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 )
2.6.39.2  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 ).
Syntax	cinl 7349	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 7350	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 7351	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inl = ( x e. _V |-> <. (/) , x >. )
Definition	df-inr 7352	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inr = ( x e. _V |-> <. 1o , x >. )
Theorem	djulclr 7353	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 ) )
Theorem	djurclr 7354	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 ) )
Theorem	djulcl 7355	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
Theorem	djurcl 7356	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
Theorem	djuf1olem 7357*	Lemma for djulf1o 7362 and djurf1o 7363. (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 )
Theorem	djuf1olemr 7358*	Lemma for djulf1or 7360 and djurf1or 7361. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7357. (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 )
Theorem	djulclb 7359	Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )
Theorem	djulf1or 7360	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 )
Theorem	djurf1or 7361	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 )
Theorem	djulf1o 7362	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 )
Theorem	djurf1o 7363	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 )
Theorem	inresflem 7364*	Lemma for inlresf1 7365 and inrresf1 7366. (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
Theorem	inlresf1 7365	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 )
Theorem	inrresf1 7366	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 )
Theorem	djuinr 7367	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 7397 and djufun 7408) 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 7389). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
|- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
Theorem	djuin 7368	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 ) ) = (/)
Theorem	inl11 7369	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 ) )
Theorem	djuunr 7370	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 )
Theorem	djuun 7371	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 )
Theorem	eldju 7372*	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 ) ) )
Theorem	djur 7373*	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 ) ) )
2.6.39.3  Universal property of the disjoint union
Theorem	djuss 7374	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
|- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
Theorem	eldju1st 7375	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 ) )
Theorem	eldju2ndl 7376	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 )
Theorem	eldju2ndr 7377	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 )
Theorem	1stinl 7378	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 ) ) = (/) )
Theorem	2ndinl 7379	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 )
Theorem	1stinr 7380	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 )
Theorem	2ndinr 7381	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 )
Theorem	djune 7382	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 ) )
Theorem	updjudhf 7383*	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 )
Theorem	updjudhcoinlf 7384*	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 )
Theorem	updjudhcoinrg 7385*	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 )
Theorem	updjud 7386*	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 ) )
Syntax	cdjucase 7387	Syntax for the "case" construction.
class case ( R , S )
Definition	df-case 7388	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 7386. 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 ) )
Theorem	casefun 7389	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 ) )
Theorem	casedm 7390	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 )
Theorem	caserel 7391	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 ) )
Theorem	casef 7392	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 )
Theorem	caseinj 7393	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 ) )
Theorem	casef1 7394	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 )
Theorem	caseinl 7395	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 ) )
Theorem	caseinr 7396	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 ) )
2.6.39.4  Dominance and equinumerosity properties of disjoint union
Theorem	djudom 7397	Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
|- ( ( A ~<_ B /\ C ~<_ D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )
Theorem	omp1eomlem 7398*	Lemma for omp1eom 7399. (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 )
Theorem	omp1eom 7399	Adding one to om. (Contributed by Jim Kingdon, 10-Jul-2023.)
|- ( om ⊔ 1o ) ~~ om
Theorem	endjusym 7400	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 ) )