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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sup00 7001 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
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Theorem | supmaxti 7002* | 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.) |
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Theorem | supsnti 7003* | The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | isotilem 7004* | Lemma for isoti 7005. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | isoti 7005* | An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | supisolem 7006* | Lemma for supisoti 7008. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | supisoex 7007* | Lemma for supisoti 7008. (Contributed by Mario Carneiro, 24-Dec-2016.) |
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Theorem | supisoti 7008* | Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.) |
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Theorem | infeq1 7009 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq1d 7010 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq1i 7011 | Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq2 7012 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq3 7013 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | infeq123d 7014 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | nfinf 7015 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
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Theorem | cnvinfex 7016* | Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | cnvti 7017* | 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.) |
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Theorem | eqinfti 7018* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
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Theorem | eqinftid 7019* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
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Theorem | infvalti 7020* | Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | infclti 7021* | An infimum belongs to its base class (closure law). See also inflbti 7022 and infglbti 7023. (Contributed by Jim Kingdon, 17-Dec-2021.) |
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Theorem | inflbti 7022* | An infimum is a lower bound. See also infclti 7021 and infglbti 7023. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infglbti 7023* | An infimum is the greatest lower bound. See also infclti 7021 and inflbti 7022. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infnlbti 7024* | A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.) |
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Theorem | infminti 7025* | 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.) |
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Theorem | infmoti 7026* |
Any class ![]() ![]() ![]() |
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Theorem | infeuti 7027* | An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | infsnti 7028* | The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | inf00 7029 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
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Theorem | infisoti 7030* | Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.) |
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Theorem | supex2g 7031 | Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
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Theorem | infex2g 7032 | Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.) |
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Theorem | ordiso2 7033 | Generalize ordiso 7034 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
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Theorem | ordiso 7034* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
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Syntax | cdju 7035 | Extend class notation to include disjoint union of two classes. |
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Definition | df-dju 7036 |
Disjoint union of two classes. This is a way of creating a class which
contains elements corresponding to each element of ![]() ![]() ![]() ![]() |
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Theorem | djueq12 7037 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | djueq1 7038 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | djueq2 7039 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | nfdju 7040 | Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
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Theorem | djuex 7041 | The disjoint union of sets is a set. See also the more precise djuss 7068. (Contributed by AV, 28-Jun-2022.) |
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Theorem | djuexb 7042 | The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
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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 | ||
Syntax | cinl 7043 | Extend class notation to include left injection of a disjoint union. |
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Syntax | cinr 7044 | Extend class notation to include right injection of a disjoint union. |
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Definition | df-inl 7045 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
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Definition | df-inr 7046 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
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Theorem | djulclr 7047 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
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Theorem | djurclr 7048 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
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Theorem | djulcl 7049 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
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Theorem | djurcl 7050 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
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Theorem | djuf1olem 7051* | Lemma for djulf1o 7056 and djurf1o 7057. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) |
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Theorem | djuf1olemr 7052* |
Lemma for djulf1or 7054 and djurf1or 7055. For a version of this lemma with
![]() ![]() |
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Theorem | djulclb 7053 | Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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Theorem | djulf1or 7054 | The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) |
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Theorem | djurf1or 7055 | The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) |
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Theorem | djulf1o 7056 | The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
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Theorem | djurf1o 7057 | The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
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Theorem | inresflem 7058* | Lemma for inlresf1 7059 and inrresf1 7060. (Contributed by BJ, 4-Jul-2022.) |
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Theorem | inlresf1 7059 | 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.) |
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Theorem | inrresf1 7060 | 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.) |
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Theorem | djuinr 7061 |
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 7091 and djufun 7102) while the simpler
statement ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | djuin 7062 | 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.) |
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Theorem | inl11 7063 | Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.) |
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Theorem | djuunr 7064 | 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.) |
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Theorem | djuun 7065 | 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.) |
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Theorem | eldju 7066* | Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.) |
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Theorem | djur 7067* | 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.) |
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Theorem | djuss 7068 | A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
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Theorem | eldju1st 7069 |
The first component of an element of a disjoint union is either ![]() ![]() |
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Theorem | eldju2ndl 7070 | 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.) |
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Theorem | eldju2ndr 7071 | 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.) |
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Theorem | 1stinl 7072 | The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
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Theorem | 2ndinl 7073 | The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.) |
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Theorem | 1stinr 7074 |
The first component of the value of a right injection is ![]() |
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Theorem | 2ndinr 7075 | The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
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Theorem | djune 7076 | Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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Theorem | updjudhf 7077* | 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.) |
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Theorem | updjudhcoinlf 7078* | 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.) |
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Theorem | updjudhcoinrg 7079* | 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.) |
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Theorem | updjud 7080* | Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.) |
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Syntax | cdjucase 7081 | Syntax for the "case" construction. |
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Definition | df-case 7082 |
The "case" construction: if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | casefun 7083 | The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | casedm 7084 |
The domain of the "case" construction is the disjoint union of the
domains. TODO (although less important):
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Theorem | caserel 7085 | 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.) |
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Theorem | casef 7086 | The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | caseinj 7087 | The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | casef1 7088 | The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | caseinl 7089 | Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.) |
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Theorem | caseinr 7090 | Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.) |
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Theorem | djudom 7091 | Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.) |
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Theorem | omp1eomlem 7092* | Lemma for omp1eom 7093. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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Theorem | omp1eom 7093 |
Adding one to ![]() |
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Theorem | endjusym 7094 | Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.) |
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Theorem | eninl 7095 | Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | eninr 7096 | Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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Theorem | difinfsnlem 7097* |
Lemma for difinfsn 7098. The case where we need to swap ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | difinfsn 7098* | An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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Theorem | difinfinf 7099* | An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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Syntax | cdjud 7100 | Syntax for the domain-disjoint-union of two relations. |
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