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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | infeq2 6901 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
inf inf | ||
Theorem | infeq3 6902 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
inf inf | ||
Theorem | infeq123d 6903 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
inf inf | ||
Theorem | nfinf 6904 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
inf | ||
Theorem | cnvinfex 6905* | Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.) |
Theorem | cnvti 6906* | 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.) |
Theorem | eqinfti 6907* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
inf | ||
Theorem | eqinftid 6908* | Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
inf | ||
Theorem | infvalti 6909* | Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.) |
inf | ||
Theorem | infclti 6910* | An infimum belongs to its base class (closure law). See also inflbti 6911 and infglbti 6912. (Contributed by Jim Kingdon, 17-Dec-2021.) |
inf | ||
Theorem | inflbti 6911* | An infimum is a lower bound. See also infclti 6910 and infglbti 6912. (Contributed by Jim Kingdon, 18-Dec-2021.) |
inf | ||
Theorem | infglbti 6912* | An infimum is the greatest lower bound. See also infclti 6910 and inflbti 6911. (Contributed by Jim Kingdon, 18-Dec-2021.) |
inf | ||
Theorem | infnlbti 6913* | A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.) |
inf | ||
Theorem | infminti 6914* | 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.) |
inf | ||
Theorem | infmoti 6915* | Any class has at most one infimum in (where is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.) |
Theorem | infeuti 6916* | An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.) |
Theorem | infsnti 6917* | The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.) |
inf | ||
Theorem | inf00 6918 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
inf | ||
Theorem | infisoti 6919* | Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.) |
inf inf | ||
Theorem | ordiso2 6920 | Generalize ordiso 6921 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Theorem | ordiso 6921* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
Syntax | cdju 6922 | Extend class notation to include disjoint union of two classes. |
⊔ | ||
Definition | df-dju 6923 | Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of or , tagging each one with whether it came from or . (Contributed by Jim Kingdon, 20-Jun-2022.) |
⊔ | ||
Theorem | djueq12 6924 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊔ ⊔ | ||
Theorem | djueq1 6925 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊔ ⊔ | ||
Theorem | djueq2 6926 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊔ ⊔ | ||
Theorem | nfdju 6927 | Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊔ | ||
Theorem | djuex 6928 | The disjoint union of sets is a set. See also the more precise djuss 6955. (Contributed by AV, 28-Jun-2022.) |
⊔ | ||
Theorem | djuexb 6929 | The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
⊔ | ||
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 and inr . | ||
Syntax | cinl 6930 | Extend class notation to include left injection of a disjoint union. |
inl | ||
Syntax | cinr 6931 | Extend class notation to include right injection of a disjoint union. |
inr | ||
Definition | df-inl 6932 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
inl | ||
Definition | df-inr 6933 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
inr | ||
Theorem | djulclr 6934 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
inl ⊔ | ||
Theorem | djurclr 6935 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
inr ⊔ | ||
Theorem | djulcl 6936 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
inl ⊔ | ||
Theorem | djurcl 6937 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
inr ⊔ | ||
Theorem | djuf1olem 6938* | Lemma for djulf1o 6943 and djurf1o 6944. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) |
Theorem | djuf1olemr 6939* | Lemma for djulf1or 6941 and djurf1or 6942. For a version of this lemma with defined on and no restriction in the conclusion, see djuf1olem 6938. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.) |
Theorem | djulclb 6940 | Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.) |
inl ⊔ | ||
Theorem | djulf1or 6941 | The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) |
inl | ||
Theorem | djurf1or 6942 | The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.) |
inr | ||
Theorem | djulf1o 6943 | The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
inl | ||
Theorem | djurf1o 6944 | The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
inr | ||
Theorem | inresflem 6945* | Lemma for inlresf1 6946 and inrresf1 6947. (Contributed by BJ, 4-Jul-2022.) |
Theorem | inlresf1 6946 | 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 ⊔ | ||
Theorem | inrresf1 6947 | 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 ⊔ | ||
Theorem | djuinr 6948 | 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 6978 and djufun 6989) while the simpler statement inl inr is easily recovered from it by substituting for both and as done in casefun 6970). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
inl inr | ||
Theorem | djuin 6949 | 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 inr | ||
Theorem | inl11 6950 | Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.) |
inl inl | ||
Theorem | djuunr 6951 | 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.) |
inl inr ⊔ | ||
Theorem | djuun 6952 | 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 inr ⊔ | ||
Theorem | eldju 6953* | Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.) |
⊔ inl inr | ||
Theorem | djur 6954* | 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.) |
⊔ inl inr | ||
Theorem | djuss 6955 | A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
⊔ | ||
Theorem | eldju1st 6956 | The first component of an element of a disjoint union is either or . (Contributed by AV, 26-Jun-2022.) |
⊔ | ||
Theorem | eldju2ndl 6957 | 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.) |
⊔ | ||
Theorem | eldju2ndr 6958 | 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.) |
⊔ | ||
Theorem | 1stinl 6959 | The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
inl | ||
Theorem | 2ndinl 6960 | The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.) |
inl | ||
Theorem | 1stinr 6961 | The first component of the value of a right injection is . (Contributed by AV, 27-Jun-2022.) |
inr | ||
Theorem | 2ndinr 6962 | The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
inr | ||
Theorem | djune 6963 | Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.) |
inl inr | ||
Theorem | updjudhf 6964* | 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.) |
⊔ ⊔ | ||
Theorem | updjudhcoinlf 6965* | 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.) |
⊔ inl | ||
Theorem | updjudhcoinrg 6966* | 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.) |
⊔ inr | ||
Theorem | updjud 6967* | Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.) |
⊔ inl inr | ||
Syntax | cdjucase 6968 | Syntax for the "case" construction. |
case | ||
Definition | df-case 6969 | The "case" construction: if and are functions, then case ⊔ 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 6967. 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 inl inr | ||
Theorem | casefun 6970 | The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
case | ||
Theorem | casedm 6971 | The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): case . (Contributed by BJ, 10-Jul-2022.) |
case ⊔ | ||
Theorem | caserel 6972 | 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 ⊔ | ||
Theorem | casef 6973 | The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.) |
case ⊔ | ||
Theorem | caseinj 6974 | The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
case | ||
Theorem | casef1 6975 | The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.) |
case ⊔ | ||
Theorem | caseinl 6976 | Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.) |
case inl | ||
Theorem | caseinr 6977 | Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.) |
case inr | ||
Theorem | djudom 6978 | Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.) |
⊔ ⊔ | ||
Theorem | omp1eomlem 6979* | Lemma for omp1eom 6980. (Contributed by Jim Kingdon, 11-Jul-2023.) |
inr inl case ⊔ | ||
Theorem | omp1eom 6980 | Adding one to . (Contributed by Jim Kingdon, 10-Jul-2023.) |
⊔ | ||
Theorem | endjusym 6981 | Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.) |
⊔ ⊔ | ||
Theorem | eninl 6982 | Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
inl | ||
Theorem | eninr 6983 | Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
inr | ||
Theorem | difinfsnlem 6984* | Lemma for difinfsn 6985. The case where we need to swap and inr in building the mapping . (Contributed by Jim Kingdon, 9-Aug-2023.) |
DECID ⊔ inr inl inr inl | ||
Theorem | difinfsn 6985* | An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
DECID | ||
Theorem | difinfinf 6986* | An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
DECID | ||
Syntax | cdjud 6987 | Syntax for the domain-disjoint-union of two relations. |
⊔d | ||
Definition | df-djud 6988 |
The "domain-disjoint-union" of two relations: if and
are two binary relations,
then ⊔d is the
binary relation from ⊔
to having the universal
property of disjoint unions (see updjud 6967 in the case of functions).
Remark: the restrictions to (resp. ) are not necessary since extra stuff would be thrown away in the post-composition with (resp. ), as in df-case 6969, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.) |
⊔d inl inr | ||
Theorem | djufun 6989 | The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
⊔d | ||
Theorem | djudm 6990 | The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.) |
⊔d ⊔ | ||
Theorem | djuinj 6991 | The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
⊔d | ||
Theorem | 0ct 6992 | The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
⊔ | ||
Theorem | ctmlemr 6993* | Lemma for ctm 6994. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
⊔ | ||
Theorem | ctm 6994* | Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
⊔ | ||
Theorem | ctssdclemn0 6995* | Lemma for ctssdc 6998. The case. (Contributed by Jim Kingdon, 16-Aug-2023.) |
DECID ⊔ | ||
Theorem | ctssdccl 6996* | A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 6998 but expressed in terms of classes rather than . (Contributed by Jim Kingdon, 30-Oct-2023.) |
⊔ inl inl DECID | ||
Theorem | ctssdclemr 6997* | Lemma for ctssdc 6998. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
⊔ DECID | ||
Theorem | ctssdc 6998* | A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7024. (Contributed by Jim Kingdon, 15-Aug-2023.) |
DECID ⊔ | ||
Theorem | enumctlemm 6999* | Lemma for enumct 7000. The case where is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.) |
Theorem | enumct 7000* | A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as ⊔ per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
⊔ |
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