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Type | Label | Description |
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Statement | ||
Theorem | rp-fakeimass 36201 | A special case where implication appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.) |
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Theorem | rp-fakeanorass 36202 | A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.) |
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Theorem | rp-fakeoranass 36203 | A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.) |
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Theorem | rp-fakenanass 36204 | A special case where nand appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.) |
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Theorem | rp-fakeinunass 36205 | A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by Richard Penner, 26-Feb-2020.) |
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Theorem | rp-fakeuninass 36206 | A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.) |
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Membership in the class of finite sets can be expressed in many ways. | ||
Theorem | rp-isfinite5 36207* |
A set is said to be finite if it can be put in one-to-one correspondence
with all the natural numbers between 1 and some ![]() ![]() ![]() |
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Theorem | rp-isfinite6 36208* |
A set is said to be finite if it is either empty or it can be put in
one-to-one correspondence with all the natural numbers between 1 and
some ![]() ![]() ![]() |
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Theorem | pwelg 36209* | The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.) |
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Theorem | pwinfig 36210* |
The powerclass of an infinite set is an infinite set, and vice-versa.
Here ![]() |
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Theorem | pwinfi2 36211 |
The powerclass of an infinite set is an infinite set, and vice-versa.
Here ![]() |
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Theorem | pwinfi3 36212 |
The powerclass of an infinite set is an infinite set, and vice-versa.
Here ![]() |
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Theorem | pwinfi 36213 | The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.) |
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While there is not yet a definition, the finite intersection property of a class is introduced by fiint 7874 where two textbook definitions are shown to be equivalent. This property is seen often with ordinal numbers (onin 5473, ordelinel 5540 ), chains of sets ordered by the proper subset relation (sorpssin 6606), various sets in the field of topology (inopn 19978, incld 20107, innei 20190, ... ) and "universal" classes like weak universes (wunin 9164, tskin 9210) and the class of all sets (inex1g 4560) . | ||
Theorem | fipjust 36214* | A definition of the finite intersection property of a class based on closure under pair-wise intersection of its elements is independent of the dummy variables. (Contributed by Richard Penner, 1-Jan-2020.) |
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Theorem | cllem0 36215* |
The class of all sets with property ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | superficl 36216* | The class of all supersets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.) |
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Theorem | superuncl 36217* | The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.) |
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Theorem | ssficl 36218* | The class of all subsets of a class has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.) |
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Theorem | ssuncl 36219* | The class of all subsets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.) |
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Theorem | ssdifcl 36220* | The class of all subsets of a class is closed under class difference. (Contributed by Richard Penner, 3-Jan-2020.) |
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Theorem | sssymdifcl 36221* | The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 3-Jan-2020.) |
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Theorem | fiinfi 36222* | If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.) |
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Theorem | elabd 36223* |
Explicit demonstration the class ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | rababg 36224 | Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.) |
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Theorem | elintabg 36225* | Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
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Theorem | elinintab 36226* | Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
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Theorem | elmapintrab 36227* | Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.) |
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Theorem | elinintrab 36228* | Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.) |
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Theorem | inintabss 36229* | Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.) |
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Theorem | inintabd 36230* | Value of the intersection of class with the intersection of a non-empty class. (Contributed by RP, 13-Aug-2020.) |
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Theorem | xpinintabd 36231* | Value of the intersection of cross-product with the intersection of a non-empty class. (Contributed by RP, 12-Aug-2020.) |
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Theorem | relintabex 36232 | If the intersection of a class is a relation, then the class is non-empty. (Contributed by RP, 12-Aug-2020.) |
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Theorem | elcnvcnvintab 36233* | Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.) |
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Theorem | relintab 36234* | Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.) |
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Theorem | nonrel 36235 | A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.) |
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Theorem | elnonrel 36236 | Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.) |
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Theorem | cnvssb 36237 | Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.) |
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Theorem | relnonrel 36238 | The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | cnvnonrel 36239 | The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.) |
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Theorem | brnonrel 36240 | A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.) |
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Theorem | dmnonrel 36241 | The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | rnnonrel 36242 | The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | resnonrel 36243 | A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | imanonrel 36244 | An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | cononrel1 36245 | Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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Theorem | cononrel2 36246 | Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.) |
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See also idssxp 28276 by Thierry Arnoux. | ||
Theorem | elmapintab 36247* |
Two ways to say a set is an element of mapped intersection of a class.
Here ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fvnonrel 36248 | The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.) |
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Theorem | elinlem 36249 | Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.) |
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Theorem | elcnvcnvlem 36250 | Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.) |
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Original probably needs new subsection for Relation-related existence theorems. | ||
Theorem | cnvcnvintabd 36251* | Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | elcnvlem 36252 | Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | elcnvintab 36253* | Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | cnvintabd 36254* | Value of the converse of the intersection of a non-empty class. (Contributed by RP, 20-Aug-2020.) |
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Theorem | undmrnresiss 36255* | Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 36256. (Contributed by RP, 26-Sep-2020.) |
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Theorem | reflexg 36256* | Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.) |
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Theorem | cnvssco 36257* | A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.) |
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Theorem | refimssco 36258 | Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.) |
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Theorem | cleq2lem 36259 | Equality implies bijection. (Contributed by RP, 24-Jul-2020.) |
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Theorem | cbvcllem 36260* | Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.) |
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Theorem | intabssd 36261* |
When for each element ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | clublem 36262* |
If a superset ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | clss2lem 36263* | The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.) |
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Theorem | dfid7 36264* | Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.) |
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Theorem | mptrcllem 36265* | Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.) |
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Theorem | cotrintab 36266 | The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.) |
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Theorem | rclexi 36267* | The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
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Theorem | rtrclexlem 36268 | Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.) |
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Theorem | rtrclex 36269* | The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.) |
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Theorem | trclubgNEW 36270* | If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.) |
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Theorem | trclubNEW 36271* | If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.) |
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Theorem | trclexi 36272* | The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
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Theorem | rtrclexi 36273* | The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.) |
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Theorem | clrellem 36274* |
When the property ![]() ![]() |
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Theorem | clcnvlem 36275* |
When ![]() |
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Theorem | cnvtrucl0 36276* | The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
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Theorem | cnvrcl0 36277* | The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
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Theorem | cnvtrcl0 36278* | The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.) |
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Theorem | dmtrcl 36279* | The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.) |
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Theorem | rntrcl 36280* | The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.) |
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Theorem | dfrtrcl5 36281* | Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.) |
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Theorem | trcleq2lemRP 36282 | Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.) |
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Theorem | al3im 36283 | Version of ax-4 1693 for a nested implication. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intima0 36284* | Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.) |
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Theorem | elimaint 36285* | Element of image of intersection. (Contributed by RP, 13-Apr-2020.) |
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Theorem | csbcog 36286 | Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.) |
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Theorem | cnviun 36287* | Converse of indexed union. (Contributed by RP, 20-Jun-2020.) |
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Theorem | imaiun1 36288* | The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.) |
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Theorem | coiun1 36289* | Composition with an indexed union. Proof analgous to that of coiun 5364. (Contributed by RP, 20-Jun-2020.) |
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Theorem | elintima 36290* | Element of intersection of images. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intimass 36291* | The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intimass2 36292* | The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intimag 36293* | Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intimasn 36294* | Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.) |
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Theorem | intimasn2 36295* | Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.) |
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Theorem | ss2iundf 36296* | Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.) |
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Theorem | ss2iundv 36297* | Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.) |
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Theorem | cbviuneq12df 36298* | Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.) |
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Theorem | cbviuneq12dv 36299* | Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.) |
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Theorem | conrel1d 36300 | Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.) |
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