P. 151 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mre1cl 15001	In any Moore collection the base set is closed. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( C e. (Moore ` X ) -> X e. C )
Theorem	mreintcl 15002	A nonempty collection of closed sets has a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> |^| S e. C )
Theorem	mreiincl 15003*	A nonempty indexed intersection of closed sets is closed. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( ( C e. (Moore ` X ) /\ I =/= (/) /\ A. y e. I S e. C ) -> |^|_ y e. I S e. C )
Theorem	mrerintcl 15004	The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( C e. (Moore ` X ) /\ S C_ C ) -> ( X i^i |^| S ) e. C )
Theorem	mreriincl 15005*	The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) -> ( X i^i |^|_ y e. I S ) e. C )
Theorem	mreincl 15006	Two closed sets have a closed intersection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( ( C e. (Moore ` X ) /\ A e. C /\ B e. C ) -> ( A i^i B ) e. C )
Theorem	mreuni 15007	Since the entire base set of a Moore collection is the greatest element of it, the base set can be recovered from a Moore collection by set union. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( C e. (Moore ` X ) -> U. C = X )
Theorem	mreunirn 15008	Two ways to express the notion of being a Moore collection on an unspecified base. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( C e. U. ran Moore <-> C e. (Moore ` U. C ) )
Theorem	ismred 15009*	Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( ph -> C C_ ~P X )   &    |- ( ph -> X e. C )   &    |- ( ( ph /\ s C_ C /\ s =/= (/) ) -> |^| s e. C )   =>    |- ( ph -> C e. (Moore ` X ) )
Theorem	ismred2 15010*	Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ph -> C C_ ~P X )   &    |- ( ( ph /\ s C_ C ) -> ( X i^i |^| s ) e. C )   =>    |- ( ph -> C e. (Moore ` X ) )
Theorem	mremre 15011	The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
|- ( X e. V -> (Moore ` X ) e. (Moore ` ~P X ) )
Theorem	submre 15012	The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ( C e. (Moore ` X ) /\ A e. C ) -> ( C i^i ~P A ) e. (Moore ` A ) )
7.2.1  Moore closures
Theorem	mrcflem 15013*	The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- ( C e. (Moore ` X ) -> ( x e. ~P X |-> |^| { s e. C | x C_ s } ) : ~P X --> C )
Theorem	fnmrc 15014	Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- mrCls Fn U. ran Moore
Theorem	mrcfval 15015*	Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) )
Theorem	mrcf 15016	The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (Moore ` X ) -> F : ~P X --> C )
Theorem	mrcval 15017*	Evaluation of the Moore closure of a set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } )
Theorem	mrccl 15018	The Moore closure of a set is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` U ) e. C )
Theorem	mrcsncl 15019	The Moore closure of a singleton is a closed set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U e. X ) -> ( F ` { U } ) e. C )
Theorem	mrcid 15020	The closure of a closed set is itself. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U e. C ) -> ( F ` U ) = U )
Theorem	mrcssv 15021	The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (Moore ` X ) -> ( F ` U ) C_ X )
Theorem	mrcidb 15022	A set is closed iff it is equal to its closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (Moore ` X ) -> ( U e. C <-> ( F ` U ) = U ) )
Theorem	mrcss 15023	Closure preserves subset ordering. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V C_ X ) -> ( F ` U ) C_ ( F ` V ) )
Theorem	mrcssid 15024	The closure of a set is a superset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> U C_ ( F ` U ) )
Theorem	mrcidb2 15025	A set is closed iff it contains its closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( U e. C <-> ( F ` U ) C_ U ) )
Theorem	mrcidm 15026	The closure operation is idempotent. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X ) -> ( F ` ( F ` U ) ) = ( F ` U ) )
Theorem	mrcsscl 15027	The closure is the minimal closed set; any closed set which contains the generators is a superset of the closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ V /\ V e. C ) -> ( F ` U ) C_ V )
Theorem	mrcuni 15028	Idempotence of closure under a general union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ ~P X ) -> ( F ` U. U ) = ( F ` U. ( F " U ) ) )
Theorem	mrcun 15029	Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (Moore ` X ) /\ U C_ X /\ V C_ X ) -> ( F ` ( U u. V ) ) = ( F ` ( ( F ` U ) u. ( F ` V ) ) ) )
Theorem	mrcssvd 15030	The Moore closure of a set is a subset of the base. Deduction form of mrcssv 15021. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   =>    |- ( ph -> ( N ` B ) C_ X )
Theorem	mrcssd 15031	Moore closure preserves subset ordering. Deduction form of mrcss 15023. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- ( ph -> U C_ V )   &    |- ( ph -> V C_ X )   =>    |- ( ph -> ( N ` U ) C_ ( N ` V ) )
Theorem	mrcssidd 15032	A set is contained in its Moore closure. Deduction form of mrcssid 15024. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- ( ph -> U C_ X )   =>    |- ( ph -> U C_ ( N ` U ) )
Theorem	mrcidmd 15033	Moore closure is idempotent. Deduction form of mrcidm 15026. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- ( ph -> U C_ X )   =>    |- ( ph -> ( N ` ( N ` U ) ) = ( N ` U ) )
Theorem	mressmrcd 15034	In a Moore system, if a set is between another set and its closure, the two sets have the same closure. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- ( ph -> S C_ ( N ` T ) )   &    |- ( ph -> T C_ S )   =>    |- ( ph -> ( N ` S ) = ( N ` T ) )
Theorem	submrc 15035	In a closure system which is cut off above some level, closures below that level act as normal. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- F = (mrCls ` C )   &    |- G = (mrCls ` ( C i^i ~P D ) )   =>    |- ( ( C e. (Moore ` X ) /\ D e. C /\ U C_ D ) -> ( G ` U ) = ( F ` U ) )
Theorem	mrieqvlemd 15036	In a Moore system, if Y is a member of S, ( S \ { Y } ) and S have the same closure if and only if Y is in the closure of ( S \ { Y } ). Used in the proof of mrieqvd 15045 and mrieqv2d 15046. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- ( ph -> S C_ X )   &    |- ( ph -> Y e. S )   =>    |- ( ph -> ( Y e. ( N ` ( S \ { Y } ) ) <-> ( N ` ( S \ { Y } ) ) = ( N ` S ) ) )
7.2.2  Independent sets in a Moore system
Theorem	mrisval 15037*	Value of the set of independent sets of a Moore system. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   =>    |- ( A e. (Moore ` X ) -> I = { s e. ~P X | A. x e. s -. x e. ( N ` ( s \ { x } ) ) } )
Theorem	ismri 15038*	Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   =>    |- ( A e. (Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
Theorem	ismri2 15039*	Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   =>    |- ( ( A e. (Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
Theorem	ismri2d 15040*	Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A e. (Moore ` X ) )   &    |- ( ph -> S C_ X )   =>    |- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
Theorem	ismri2dd 15041*	Definition of independence of a subset of the base set in a Moore system. One-way deduction form. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A e. (Moore ` X ) )   &    |- ( ph -> S C_ X )   &    |- ( ph -> A. x e. S -. x e. ( N ` ( S \ { x } ) ) )   =>    |- ( ph -> S e. I )
Theorem	mriss 15042	An independent set of a Moore system is a subset of the base set. (Contributed by David Moews, 1-May-2017.)
|- I = (mrInd ` A )   =>    |- ( ( A e. (Moore ` X ) /\ S e. I ) -> S C_ X )
Theorem	mrissd 15043	An independent set of a Moore system is a subset of the base set. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- I = (mrInd ` A )   &    |- ( ph -> A e. (Moore ` X ) )   &    |- ( ph -> S e. I )   =>    |- ( ph -> S C_ X )
Theorem	ismri2dad 15044	Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017.)
|- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A e. (Moore ` X ) )   &    |- ( ph -> S e. I )   &    |- ( ph -> Y e. S )   =>    |- ( ph -> -. Y e. ( N ` ( S \ { Y } ) ) )
Theorem	mrieqvd 15045*	In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> S C_ X )   =>    |- ( ph -> ( S e. I <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )
Theorem	mrieqv2d 15046*	In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> S C_ X )   =>    |- ( ph -> ( S e. I <-> A. s ( s C. S -> ( N ` s ) C. ( N ` S ) ) ) )
Theorem	mrissmrcd 15047	In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 15034, and so are equal by mrieqv2d 15046.) (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> S C_ ( N ` T ) )   &    |- ( ph -> T C_ S )   &    |- ( ph -> S e. I )   =>    |- ( ph -> S = T )
Theorem	mrissmrid 15048	In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> S e. I )   &    |- ( ph -> T C_ S )   =>    |- ( ph -> T e. I )
Theorem	mreexd 15049*	In a Moore system, the closure operator is said to have the exchange property if, for all elements y and z of the base set and subsets S of the base set such that z is in the closure of ( S u. { y } ) but not in the closure of S, y is in the closure of ( S u. { z } ) (Definition 3.1.9 in [FaureFrolicher] p. 57 to 58.) This theorem allows us to construct substitution instances of this definition. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> X e. V )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S C_ X )   &    |- ( ph -> Y e. X )   &    |- ( ph -> Z e. ( N ` ( S u. { Y } ) ) )   &    |- ( ph -> -. Z e. ( N ` S ) )   =>    |- ( ph -> Y e. ( N ` ( S u. { Z } ) ) )
Theorem	mreexmrid 15050*	In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S e. I )   &    |- ( ph -> Y e. X )   &    |- ( ph -> -. Y e. ( N ` S ) )   =>    |- ( ph -> ( S u. { Y } ) e. I )
Theorem	mreexexlemd 15051*	This lemma is used to generate substitution instances of the induction hypothesis in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> X e. J )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F ~~ K \/ G ~~ K ) )   &    |- ( ph -> A. t A. u e. ~P ( X \ t ) A. v e. ~P ( X \ t ) ( ( ( u ~~ K \/ v ~~ K ) /\ u C_ ( N ` ( v u. t ) ) /\ ( u u. t ) e. I ) -> E. i e. ~P v ( u ~~ i /\ ( i u. t ) e. I ) ) )   =>    |- ( ph -> E. j e. ~P G ( F ~~ j /\ ( j u. H ) e. I ) )
Theorem	mreexexlem2d 15052*	Used in mreexexlem4d 15054 to prove the induction step in mreexexd 15055. See the proof of Proposition 4.2.1 in [FaureFrolicher] p. 86 to 87. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> Y e. F )   =>    |- ( ph -> E. g e. G ( -. g e. ( F \ { Y } ) /\ ( ( F \ { Y } ) u. ( H u. { g } ) ) e. I ) )
Theorem	mreexexlem3d 15053*	Base case of the induction in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F = (/) \/ G = (/) ) )   =>    |- ( ph -> E. i e. ~P G ( F ~~ i /\ ( i u. H ) e. I ) )
Theorem	mreexexlem4d 15054*	Induction step of the induction in mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> L e. om )   &    |- ( ph -> A. h A. f e. ~P ( X \ h ) A. g e. ~P ( X \ h ) ( ( ( f ~~ L \/ g ~~ L ) /\ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) -> E. j e. ~P g ( f ~~ j /\ ( j u. h ) e. I ) ) )   &    |- ( ph -> ( F ~~ suc L \/ G ~~ suc L ) )   =>    |- ( ph -> E. j e. ~P G ( F ~~ j /\ ( j u. H ) e. I ) )
Theorem	mreexexd 15055*	Exchange-type theorem. In a Moore system whose closure operator has the exchange property, if F and G are disjoint from H, ( F u. H ) is independent, F is contained in the closure of ( G u. H ), and either F or G is finite, then there is a subset q of G equinumerous to F such that ( q u. H ) is independent. This implies the case of Proposition 4.2.1 in [FaureFrolicher] p. 86 where either ( A \ B ) or ( B \ A ) is finite. The theorem is proven by induction using mreexexlem3d 15053 for the base case and mreexexlem4d 15054 for the induction step. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> F C_ ( X \ H ) )   &    |- ( ph -> G C_ ( X \ H ) )   &    |- ( ph -> F C_ ( N ` ( G u. H ) ) )   &    |- ( ph -> ( F u. H ) e. I )   &    |- ( ph -> ( F e. Fin \/ G e. Fin ) )   =>    |- ( ph -> E. q e. ~P G ( F ~~ q /\ ( q u. H ) e. I ) )
Theorem	mreexdomd 15056*	In a Moore system whose closure operator has the exchange property, if S is independent and contained in the closure of T, and either S or T is finite, then T dominates S. This is an immediate consequence of mreexexd 15055. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S C_ ( N ` T ) )   &    |- ( ph -> T C_ X )   &    |- ( ph -> ( S e. Fin \/ T e. Fin ) )   &    |- ( ph -> S e. I )   =>    |- ( ph -> S ~<_ T )
Theorem	mreexfidimd 15057*	In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 15056 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (Moore ` X ) )   &    |- N = (mrCls ` A )   &    |- I = (mrInd ` A )   &    |- ( ph -> A. s e. ~P X A. y e. X A. z e. ( ( N ` ( s u. { y } ) ) \ ( N ` s ) ) y e. ( N ` ( s u. { z } ) ) )   &    |- ( ph -> S e. I )   &    |- ( ph -> T e. I )   &    |- ( ph -> S e. Fin )   &    |- ( ph -> ( N ` S ) = ( N ` T ) )   =>    |- ( ph -> S ~~ T )
7.2.3  Algebraic closure systems
Theorem	isacs 15058*	A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ E. f ( f : ~P X --> ~P X /\ A. s e. ~P X ( s e. C <-> U. ( f " ( ~P s i^i Fin ) ) C_ s ) ) ) )
Theorem	acsmre 15059	Algebraic closure systems are closure systems. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- ( C e. (ACS ` X ) -> C e. (Moore ` X ) )
Theorem	isacs2 15060*	In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (ACS ` X ) <-> ( C e. (Moore ` X ) /\ A. s e. ~P X ( s e. C <-> A. y e. ( ~P s i^i Fin ) ( F ` y ) C_ s ) ) )
Theorem	acsfiel 15061*	A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( C e. (ACS ` X ) -> ( S e. C <-> ( S C_ X /\ A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) ) )
Theorem	acsfiel2 15062*	A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- F = (mrCls ` C )   =>    |- ( ( C e. (ACS ` X ) /\ S C_ X ) -> ( S e. C <-> A. y e. ( ~P S i^i Fin ) ( F ` y ) C_ S ) )
Theorem	acsmred 15063	An algebraic closure system is also a Moore system. Deduction form of acsmre 15059. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. (ACS ` X ) )   =>    |- ( ph -> A e. (Moore ` X ) )
Theorem	isacs1i 15064*	A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ F : ~P X --> ~P X ) -> { s e. ~P X | U. ( F " ( ~P s i^i Fin ) ) C_ s } e. (ACS ` X ) )
Theorem	mreacs 15065	Algebraicity is a composable property; combining several algebraic closure properties gives another. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( X e. V -> (ACS ` X ) e. (Moore ` ~P X ) )
Theorem	acsfn 15066*	Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( ( X e. V /\ K e. X ) /\ ( T C_ X /\ T e. Fin ) ) -> { a e. ~P X | ( T C_ a -> K e. a ) } e. (ACS ` X ) )
Theorem	acsfn0 15067*	Algebraicity of a point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ K e. X ) -> { a e. ~P X | K e. a } e. (ACS ` X ) )
Theorem	acsfn1 15068*	Algebraicity of a one-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. X E e. X ) -> { a e. ~P X | A. b e. a E e. a } e. (ACS ` X ) )
Theorem	acsfn1c 15069*	Algebraicity of a one-argument closure condition with additional constant. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. K A. c e. X E e. X ) -> { a e. ~P X | A. b e. K A. c e. a E e. a } e. (ACS ` X ) )
Theorem	acsfn2 15070*	Algebraicity of a two-argument closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
|- ( ( X e. V /\ A. b e. X A. c e. X E e. X ) -> { a e. ~P X | A. b e. a A. c e. a E e. a } e. (ACS ` X ) )
PART 8  BASIC CATEGORY THEORY
8.1  Categories
8.1.1  Categories
Syntax	ccat 15071	Extend class notation with the class of categories.
class Cat
Syntax	ccid 15072	Extend class notation with the identity arrow of a category.
class Id
Syntax	chomf 15073	Extend class notation to include functionalized Hom-set extractor.
class Hom f
Syntax	ccomf 15074	Extend class notation to include functionalized composition operation.
class compf
Definition	df-cat 15075*	A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. Definition in [Lang] p. 53. In contrast to definition 3.1 of [Adamek] p. 21, where "A category is a quadruple A = (O, hom, id, o)", a category is defined as an extensible structure consisting of three slots: the objects "O" ( ( Base ` c )), the morphisms "hom" ( ( Hom ` c )) and the composition law "o" ( (comp ` c )). The identities "id" are defined by their properties related to morphisms and their composition, see condition 3.1(b) in [Adamek] p. 21 and df-cid 15076. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 2-Jan-2017.)
|- Cat = { c | [. ( Base ` c ) / b ]. [. ( Hom ` c ) / h ]. [. (comp ` c ) / o ]. A. x e. b ( E. g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) /\ A. y e. b A. z e. b A. f e. ( x h y ) A. g e. ( y h z ) ( ( g ( <. x , y >. o z ) f ) e. ( x h z ) /\ A. w e. b A. k e. ( z h w ) ( ( k ( <. y , z >. o w ) g ) ( <. x , y >. o w ) f ) = ( k ( <. x , z >. o w ) ( g ( <. x , y >. o z ) f ) ) ) ) }
Definition	df-cid 15076*	Define the category identity arrow. Since it is uniquely defined when it exists, we do not need to add it to the data of the category, and instead extract it by uniqueness. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- Id = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ [_ (comp ` c ) / o ]_ ( x e. b |-> ( iota_ g e. ( x h x ) A. y e. b ( A. f e. ( y h x ) ( g ( <. y , x >. o x ) f ) = f /\ A. f e. ( x h y ) ( f ( <. x , x >. o y ) g ) = f ) ) ) )
Definition	df-homf 15077*	Define the functionalized Hom-set operator, which is exactly like Hom but is guaranteed to be a function on the base. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- Hom f = ( c e. _V |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( x ( Hom ` c ) y ) ) )
Definition	df-comf 15078*	Define the functionalized composition operator, which is exactly like comp but is guaranteed to be a function of the proper type. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- compf = ( c e. _V |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) , y e. ( Base ` c ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` c ) y ) , f e. ( ( Hom ` c ) ` x ) |-> ( g ( x (comp ` c ) y ) f ) ) ) )
Theorem	iscat 15079*	The predicate "is a category". (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   =>    |- ( C e. V -> ( C e. Cat <-> A. x e. B ( E. g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) /\ A. y e. B A. z e. B A. f e. ( x H y ) A. g e. ( y H z ) ( ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) /\ A. w e. B A. k e. ( z H w ) ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) ) ) ) )
Theorem	iscatd 15080*	Properties that determine a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. V )   &    |- ( ( ph /\ x e. B ) -> .1. e. ( x H x ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( y H x ) ) ) -> ( .1. ( <. y , x >. .x. x ) f ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( x H y ) ) ) -> ( f ( <. x , x >. .x. y ) .1. ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) ) ) -> ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) )   &    |- ( ( ph /\ ( ( x e. B /\ y e. B ) /\ ( z e. B /\ w e. B ) ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) /\ k e. ( z H w ) ) ) -> ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) )   =>    |- ( ph -> C e. Cat )
Theorem	catidex 15081*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> E. g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) )
Theorem	catideu 15082*	Each object in a category has a unique identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> E! g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) )
Theorem	cidfval 15083*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   =>    |- ( ph -> .1. = ( x e. B |-> ( iota_ g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) ) ) )
Theorem	cidval 15084*	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- .1. = ( Id ` C )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .1. ` X ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )
Theorem	cidffn 15085	The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
|- Id Fn Cat
Theorem	cidfn 15086	The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- B = ( Base ` C )   &    |- .1. = ( Id ` C )   =>    |- ( C e. Cat -> .1. Fn B )
Theorem	catidd 15087*	Deduce the identity arrow in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. Cat )   &    |- ( ( ph /\ x e. B ) -> .1. e. ( x H x ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( y H x ) ) ) -> ( .1. ( <. y , x >. .x. x ) f ) = f )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ f e. ( x H y ) ) ) -> ( f ( <. x , x >. .x. y ) .1. ) = f )   =>    |- ( ph -> ( Id ` C ) = ( x e. B |-> .1. ) )
Theorem	iscatd2 15088*	Version of iscatd 15080 with a uniform assumption list, for increased proof sharing capabilities. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> H = ( Hom ` C ) )   &    |- ( ph -> .x. = (comp ` C ) )   &    |- ( ph -> C e. V )   &    |- ( ps <-> ( ( x e. B /\ y e. B ) /\ ( z e. B /\ w e. B ) /\ ( f e. ( x H y ) /\ g e. ( y H z ) /\ k e. ( z H w ) ) ) )   &    |- ( ( ph /\ y e. B ) -> .1. e. ( y H y ) )   &    |- ( ( ph /\ ps ) -> ( .1. ( <. x , y >. .x. y ) f ) = f )   &    |- ( ( ph /\ ps ) -> ( g ( <. y , y >. .x. z ) .1. ) = g )   &    |- ( ( ph /\ ps ) -> ( g ( <. x , y >. .x. z ) f ) e. ( x H z ) )   &    |- ( ( ph /\ ps ) -> ( ( k ( <. y , z >. .x. w ) g ) ( <. x , y >. .x. w ) f ) = ( k ( <. x , z >. .x. w ) ( g ( <. x , y >. .x. z ) f ) ) )   =>    |- ( ph -> ( C e. Cat /\ ( Id ` C ) = ( y e. B |-> .1. ) ) )
Theorem	catidcl 15089	Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .1. ` X ) e. ( X H X ) )
Theorem	catlid 15090	Left identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- .x. = (comp ` C )   &    |- ( ph -> Y e. B )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( ( .1. ` Y ) ( <. X , Y >. .x. Y ) F ) = F )
Theorem	catrid 15091	Right identity property of an identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .1. = ( Id ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- .x. = (comp ` C )   &    |- ( ph -> Y e. B )   &    |- ( ph -> F e. ( X H Y ) )   =>    |- ( ph -> ( F ( <. X , X >. .x. Y ) ( .1. ` X ) ) = F )
Theorem	catcocl 15092	Closure of a composition arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   =>    |- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) e. ( X H Z ) )
Theorem	catass 15093	Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- .x. = (comp ` C )   &    |- ( ph -> C e. Cat )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F e. ( X H Y ) )   &    |- ( ph -> G e. ( Y H Z ) )   &    |- ( ph -> W e. B )   &    |- ( ph -> K e. ( Z H W ) )   =>    |- ( ph -> ( ( K ( <. Y , Z >. .x. W ) G ) ( <. X , Y >. .x. W ) F ) = ( K ( <. X , Z >. .x. W ) ( G ( <. X , Y >. .x. Z ) F ) ) )
Theorem	0catg 15094	Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- ( ( C e. V /\ (/) = ( Base ` C ) ) -> C e. Cat )
Theorem	0cat 15095	The empty set is a category, the empty category, see example 3.3(4.c) in [Adamek] p. 24. (Contributed by Mario Carneiro, 3-Jan-2017.)
|- (/) e. Cat
Theorem	homffval 15096*	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   =>    |- F = ( x e. B , y e. B |-> ( x H y ) )
Theorem	fnhomeqhomf 15097	If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   =>    |- ( H Fn ( B X. B ) -> F = H )
Theorem	homfval 15098	Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   &    |- H = ( Hom ` C )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X F Y ) = ( X H Y ) )
Theorem	homffn 15099	The functionalized Hom-set operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
|- F = ( Hom f ` C )   &    |- B = ( Base ` C )   =>    |- F Fn ( B X. B )
Theorem	homfeq 15100*	Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
|- H = ( Hom ` C )   &    |- J = ( Hom ` D )   &    |- ( ph -> B = ( Base ` C ) )   &    |- ( ph -> B = ( Base ` D ) )   =>    |- ( ph -> ( ( Hom f ` C ) = ( Hom f ` D ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) ) )