P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	unfiin 7201	The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) e. Fin ) -> ( A u. B ) e. Fin )
Theorem	prfidisj 7202	A pair is finite if it consists of two unequal sets. For the case where A = B, see snfig 7071. For the cases where one or both is a proper class, see prprc1 3806, prprc2 3807, or prprc 3808. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin )
Theorem	prfidceq 7203*	A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> A. x e. C A. y e. C DECID x = y )   =>    |- ( ph -> { A , B } e. Fin )
Theorem	tpfidisj 7204	A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> { A , B , C } e. Fin )
Theorem	tpfidceq 7205*	A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. D )   &    |- ( ph -> B e. D )   &    |- ( ph -> C e. D )   &    |- ( ph -> A. x e. D A. y e. D DECID x = y )   =>    |- ( ph -> { A , B , C } e. Fin )
Theorem	fiintim 7206*	If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as x and y not being equal, or A having decidable equality. This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( A. x e. A A. y e. A ( x i^i y ) e. A -> A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) )
Theorem	xpfi 7207	The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin )
Theorem	imaf1fi 7208	The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
|- ( ( F : A -1-1-> B /\ X C_ A /\ X e. Fin ) -> ( F " X ) e. Fin )
Theorem	3xpfi 7209	The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
|- ( V e. Fin -> ( ( V X. V ) X. V ) e. Fin )
Theorem	fisseneq 7210	A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
|- ( ( B e. Fin /\ A C_ B /\ A ~~ B ) -> A = B )
Theorem	phpeqd 7211	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 7135 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B C_ A )   &    |- ( ph -> A ~~ B )   =>    |- ( ph -> A = B )
Theorem	ssfirab 7212*	A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A. x e. A DECID ps )   =>    |- ( ph -> { x e. A | ps } e. Fin )
Theorem	ssfidc 7213*	A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
|- ( ( A e. Fin /\ B C_ A /\ A. x e. A DECID x e. B ) -> B e. Fin )
Theorem	exmidssfi 7214*	Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
|- (EXMID <-> A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) )
Theorem	opabfi 7215*	Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> A. x e. A A. y e. B DECID ps )   =>    |- ( ph -> S e. Fin )
Theorem	infidc 7216*	The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ( A e. Fin /\ A. x e. A DECID x e. B ) -> ( A i^i B ) e. Fin )
Theorem	snon0 7217	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 7218	A version of fnex 5913 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
Theorem	fundmfi 7219	The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( A e. Fin /\ Fun A ) -> dom A e. Fin )
Theorem	fundmfibi 7220	A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
|- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) )
Theorem	resfnfinfinss 7221	The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
|- ( ( F Fn A /\ B e. Fin /\ B C_ A ) -> ( F |` B ) e. Fin )
Theorem	residfi 7222	A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
|- ( ( _I |` A ) e. Fin <-> A e. Fin )
Theorem	relcnvfi 7223	If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
Theorem	funrnfi 7224	The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )
Theorem	f1ofi 7225	If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> B e. Fin )
Theorem	f1dmvrnfibi 7226	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7227. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 7227	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7226. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 7228*	The finite union of disjoint finite sets is finite. Note that B depends on x, i.e. can be thought of as B ( x ). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
|- ( ( A e. Fin /\ A. x e. A B e. Fin /\ Disj x e. A B ) -> U_ x e. A B e. Fin )
Theorem	mapfi 7229	Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A ^m B ) e. Fin )
Theorem	elfpw 7230	Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( A e. ( ~P B i^i Fin ) <-> ( A C_ B /\ A e. Fin ) )
Theorem	fissfi 7231*	A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
Theorem	f1finf1o 7232	Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
|- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )
Theorem	en1eqsn 7233	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 7234	A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
|- ( A e. B -> ( B ~~ 1o <-> B = { A } ) )
Theorem	snexxph 7235*	A case where the antecedent of snexg 4303 is not needed. The class { x | ph } is from dcextest 4710. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 7236	The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
|- ( ph -> C C_ B )   &    |- ( ph -> F : A -1-1-onto-> B )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> ( `' F " C ) e. Fin )
Theorem	fidcenumlemim 7237*	Lemma for fidcenum 7241. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( A e. Fin -> ( A. x e. A A. y e. A DECID x = y /\ E. n e. om E. f f : n -onto-> A ) )
Theorem	fidcenumlemrks 7238*	Lemma for fidcenum 7241. Induction step for fidcenumlemrk 7239. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> J e. om )   &    |- ( ph -> suc J C_ N )   &    |- ( ph -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
Theorem	fidcenumlemrk 7239*	Lemma for fidcenum 7241. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> K C_ N )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) )
Theorem	fidcenumlemr 7240*	Lemma for fidcenum 7241. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   =>    |- ( ph -> A e. Fin )
Theorem	fidcenum 7241*	A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as E. n e. om E. f f : n -onto-> A is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( A e. Fin <-> ( A. x e. A A. y e. A DECID x = y /\ E. n e. om E. f f : n -onto-> A ) )
2.6.33  Schroeder-Bernstein Theorem
Theorem	sbthlem1 7242*	Lemma for isbth 7252. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
Theorem	sbthlem2 7243*	Lemma for isbth 7252. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
Theorem	sbthlemi3 7244*	Lemma for isbth 7252. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
Theorem	sbthlemi4 7245*	Lemma for isbth 7252. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
Theorem	sbthlemi5 7246*	Lemma for isbth 7252. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ ( dom f = A /\ ran g C_ A ) ) -> dom H = A )
Theorem	sbthlemi6 7247*	Lemma for isbth 7252. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
Theorem	sbthlem7 7248*	Lemma for isbth 7252. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( Fun f /\ Fun `' g ) -> Fun H )
Theorem	sbthlemi8 7249*	Lemma for isbth 7252. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
Theorem	sbthlemi9 7250*	Lemma for isbth 7252. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
Theorem	sbthlemi10 7251*	Lemma for isbth 7252. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   &    |- B e. _V   =>    |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Theorem	isbth 7252	Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set A is smaller (has lower cardinality) than B and vice-versa, then A and B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7242 through sbthlemi10 7251; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7251. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 16945. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.34  Finitely supported functions
Syntax	cfsupp 7253	Extend class definition to include the predicate to be a finitely supported function.
class finSupp
Definition	df-fsupp 7254*	Define the property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
|- finSupp = { <. r , z >. | ( Fun r /\ ( r supp z ) e. Fin ) }
Theorem	relfsupp 7255	The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
|- Rel finSupp
Theorem	relprcnfsupp 7256	A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
|- ( -. A e. _V -> -. A finSupp Z )
Theorem	isfsupp 7257	The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
|- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
Theorem	isfsuppd 7258	Deduction form of isfsupp 7257. (Contributed by SN, 29-Jul-2024.)
|- ( ph -> R e. V )   &    |- ( ph -> Z e. W )   &    |- ( ph -> Fun R )   &    |- ( ph -> ( R supp Z ) e. Fin )   =>    |- ( ph -> R finSupp Z )
Theorem	funisfsupp 7259	The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)
|- ( ( Fun R /\ R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( R supp Z ) e. Fin ) )
Theorem	fsuppimp 7260	Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)
|- ( R finSupp Z -> ( Fun R /\ ( R supp Z ) e. Fin ) )
Theorem	fsuppimpd 7261	A finitely supported function is a function with a finite support. (Contributed by AV, 6-Jun-2019.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> ( F supp Z ) e. Fin )
Theorem	fsuppfund 7262	A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> Fun F )
Theorem	suppeqfsuppbi 7263	If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)
|- ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( ( F supp Z ) = ( G supp Z ) -> ( F finSupp Z <-> G finSupp Z ) ) )
Theorem	fsuppxpfi 7264	The cartesian product of two finitely supported functions is finite. (Contributed by AV, 17-Jul-2019.)
|- ( ( F finSupp Z /\ G finSupp Z ) -> ( ( F supp Z ) X. ( G supp Z ) ) e. Fin )
Theorem	fczfsuppd 7265	A constant function with value zero is finitely supported. (Contributed by AV, 30-Jun-2019.)
|- ( ph -> B e. V )   &    |- ( ph -> Z e. W )   =>    |- ( ph -> ( B X. { Z } ) finSupp Z )
Theorem	0fsupp 7266	The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
|- ( Z e. V -> (/) finSupp Z )
Theorem	snopfsuppdc 7267	A singleton containing an ordered pair is a finitely supported function. (Contributed by AV, 19-Jul-2019.)
|- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> Z e. U )   &    |- ( ph -> DECID Y = Z )   =>    |- ( ph -> { <. X , Y >. } finSupp Z )
Theorem	ffsuppbi 7268	Two ways of saying that a function with known codomain is finitely supported. (Contributed by AV, 8-Jul-2019.)
|- ( ( I e. V /\ Z e. W ) -> ( F : I --> S -> ( F finSupp Z <-> ( `' F " ( S \ { Z } ) ) e. Fin ) ) )
Theorem	fsuppcorn 7269	The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the ( F supp Z ) C_ ran G condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7214. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)
|- ( ph -> F finSupp Z )   &    |- ( ph -> G : X -1-1-> Y )   &    |- ( ph -> Z e. W )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. U )   &    |- ( ph -> ( F supp Z ) C_ ran G )   =>    |- ( ph -> ( F o. G ) finSupp Z )
2.6.35  Finite intersections
Syntax	cfi 7270	Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
Definition	df-fi 7271*	Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 7274). (Contributed by FL, 27-Apr-2008.)
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
Theorem	fival 7272*	The set of all the finite intersections of the elements of A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> ( fi ` A ) = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )
Theorem	elfi 7273*	Specific properties of an element of ( fi ` B ). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( ( A e. V /\ B e. W ) -> ( A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )
Theorem	elfi2 7274*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( B e. V -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )
Theorem	elfir 7275	Sufficient condition for an element of ( fi ` B ). (Contributed by Mario Carneiro, 24-Nov-2013.)
|- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> |^| A e. ( fi ` B ) )
Theorem	ssfii 7276	Any element of a set A is the intersection of a finite subset of A. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( A e. V -> A C_ ( fi ` A ) )
Theorem	fi0 7277	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( fi ` (/) ) = (/)
Theorem	fieq0 7278	A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> ( A = (/) <-> ( fi ` A ) = (/) ) )
Theorem	fiss 7279	Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )
Theorem	fiuni 7280	The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> U. A = U. ( fi ` A ) )
Theorem	fipwssg 7281	If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( A e. V /\ A C_ ~P X ) -> ( fi ` A ) C_ ~P X )
Theorem	fifo 7282*	Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
|- F = ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| y )   =>    |- ( A e. V -> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )
Theorem	dcfi 7283*	Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( ( A e. Fin /\ A. x e. A DECID ph ) -> DECID A. x e. A ph )
2.6.36  The sizes of sets
Theorem	2omap 7284*	Mapping between ( 2o ^m A ) and decidable subsets of A. (Contributed by Jim Kingdon, 12-Nov-2025.)
|- F = ( s e. ( 2o ^m A ) |-> { z e. A | ( s ` z ) = 1o } )   =>    |- ( A e. V -> F : ( 2o ^m A ) -1-1-onto-> { x e. ~P A | A. y e. A DECID y e. x } )
Theorem	2omapen 7285*	Equinumerosity of ( 2o ^m A ) and the set of decidable subsets of A. (Contributed by Jim Kingdon, 14-Nov-2025.)
|- ( A e. V -> ( 2o ^m A ) ~~ { x e. ~P A | A. y e. A DECID y e. x } )
Theorem	2omapfi 7286	The number of finite subsets of a finite set. For a similar theorem with set size expressed using ♯ (df-ihash 11169), see hashpwfi 11223. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( A e. Fin -> ( 2o ^m A ) ~~ ( ~P A i^i Fin ) )
Theorem	fipwfi 7287	The set of finite subsets of a finite set is finite. (Contributed by Jim Kingdon, 19-May-2026.)
|- ( A e. Fin -> ( ~P A i^i Fin ) e. Fin )
2.6.37  Supremum and infimum
Syntax	csup 7288	Extend class notation to include supremum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class sup ( A , B , R )
Syntax	cinf 7289	Extend class notation to include infimum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class inf ( A , B , R )
Definition	df-sup 7290*	Define the supremum of class A. It is meaningful when R is a relation that strictly orders B and when the supremum exists. (Contributed by NM, 22-May-1999.)
|- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\ A. y e. B ( y R x -> E. z e. A y R z ) ) }
Definition	df-inf 7291	Define the infimum of class A. It is meaningful when R is a relation that strictly orders B and when the infimum exists. For example, R could be 'less than', B could be the set of real numbers, and A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
|- inf ( A , B , R ) = sup ( A , B , `' R )
Theorem	supeq1 7292	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
|- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1d 7293	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> B = C )   =>    |- ( ph -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1i 7294	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- B = C   =>    |- sup ( B , A , R ) = sup ( C , A , R )
Theorem	supeq2 7295	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) )
Theorem	supeq3 7296	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
|- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) )
Theorem	supeq123d 7297	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> sup ( A , B , C ) = sup ( D , E , F ) )
Theorem	nfsup 7298	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ x sup ( A , B , R )
Theorem	supmoti 7299*	Any class B has at most one supremum in A (where R is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8371) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-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 ) ) )
Theorem	supeuti 7300*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-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 -> 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 ) ) )