P. 371 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	adantlllr 37001	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ( ( ph /\ et ) /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	3adantlr3 37002	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ( ph /\ ( ps /\ ch ) ) /\ th ) -> ta )   =>    |- ( ( ( ph /\ ( ps /\ ch /\ et ) ) /\ th ) -> ta )
Theorem	nnxrd 37003	A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR* )
Theorem	3adantll2 37004	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ( ph /\ et /\ ps ) /\ ch ) /\ th ) -> ta )
Theorem	3adantll3 37005	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )   =>    |- ( ( ( ( ph /\ ps /\ et ) /\ ch ) /\ th ) -> ta )
Theorem	ssnel 37006	If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A C_ B /\ -. C e. B ) -> -. C e. A )
Theorem	adant423 37007	Deduction adding conjuncts to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) -> ch )
Theorem	jcn 37008	Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ps )   &    |- ( ph -> -. ch )   =>    |- ( ph -> -. ( ps -> ch ) )
Theorem	3expc 37009	Exportation inference. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( ( ph /\ ps ) /\ ch ) -> th )   =>    |- ( ph -> ( ps -> ( ch -> th ) ) )
Theorem	elabrexg 37010*	Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( x e. A /\ B e. V ) -> B e. { y | E. x e. A y = B } )
Theorem	unicntop 37011	The underlying set of the standard topology on the complex numers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- CC = U. ( TopOpen ` ℂfld )
Theorem	ifeq123d 37012	Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 3942. TODO (NM): Please replace the usage of this theorem by ifbieq12d 3942 then delete this theorem. (New usage is discouraged.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> if ( ps , A , C ) = if ( ch , B , D ) )
Theorem	sncldre 37013	A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR -> { A } e. ( Clsd ` ( topGen ` ran (,) ) ) )
Theorem	neqne 37014	From non equality to inequality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( -. A = B -> A =/= B )
Theorem	cnopn 37015	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- CC e. ( TopOpen ` ℂfld )
Theorem	n0p 37016	A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ( P e. (Poly ` ZZ ) /\ N e. NN0 /\ ( (coeff ` P ) ` N ) =/= 0 ) -> P =/= 0p )
Theorem	rabeqd 37017*	Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> { x e. A | ps } = { x e. B | ps } )
Theorem	pm2.65ni 37018	Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( -. ph -> ps )   &    |- ( -. ph -> -. ps )   =>    |- ph
Theorem	raleqdOLD 37019*	Equality theorem for restricted universal quantifier. (Contributed by Glauco Siliprandi, 5-Apr-2020.) Obsolete as of 18-Jul-2020. Use raleqdv 3038 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
Theorem	elini 37020	Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- A e. B   &    |- A e. C   =>    |- A e. ( B i^i C )
Theorem	pwssfi 37021	Every element of the power set of A is finite if and only if A is finite (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A e. V -> ( A e. Fin <-> ~P A C_ Fin ) )
Theorem	iuneq2df 37022	Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B = C )   =>    |- ( ph -> U_ x e. A B = U_ x e. A C )
Theorem	nnfoctb 37023*	There exists a mapping from NN onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( A ~<_ om /\ A =/= (/) ) -> E. f f : NN -onto-> A )
Theorem	prssd 37024	A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   =>    |- ( ph -> { A , B } C_ C )
Theorem	ssinss1d 37025	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A C_ C )   =>    |- ( ph -> ( A i^i B ) C_ C )
Theorem	prcnel 37026	A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( -. A e. _V -> -. A e. V )
Theorem	0un 37027	The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( (/) u. A ) = A
Theorem	elpwinss 37028	An element of the powerset of B intersected with anything, is a subset of B. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A e. ( ~P B i^i C ) -> A C_ B )
Theorem	unidmex 37029	If F is a set, then U. dom F is a set (common case) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F e. V )   &    |- X = U. dom F   =>    |- ( ph -> X e. _V )
Theorem	ndisj2 37030*	A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( x = y -> B = C )   =>    |- ( -. Disj x e. A B <-> E. x e. A E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )
Theorem	zenom 37031	The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ZZ ~~ om
Theorem	rexsngf 37032*	Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ps   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
Theorem	uzwo4 37033*	Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ j ps   &    |- ( j = k -> ( ph <-> ps ) )   =>    |- ( ( S C_ ( ZZ>= ` M ) /\ E. j e. S ph ) -> E. j e. S ( ph /\ A. k e. S ( k < j -> -. ps ) ) )
Theorem	0in 37034	The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( (/) i^i A ) = (/)
Theorem	unisn0 37035	The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- U. { (/) } = (/)
Theorem	ssin0 37036	If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ( A i^i B ) = (/) /\ C C_ A /\ D C_ B ) -> ( C i^i D ) = (/) )
Theorem	inabs3 37037	Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( C C_ B -> ( ( A i^i B ) i^i C ) = ( A i^i C ) )
Theorem	pwpwuni 37038	Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A e. V -> ( A e. ~P ~P B <-> U. A e. ~P B ) )
Theorem	disjiun2 37039*	In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> Disj x e. A B )   &    |- ( ph -> C C_ A )   &    |- ( ph -> D e. ( A \ C ) )   &    |- ( x = D -> B = E )   =>    |- ( ph -> ( U_ x e. C B i^i E ) = (/) )
Theorem	0pwfi 37040	The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- (/) e. ( ~P A i^i Fin )
Theorem	ssinss2d 37041	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> B C_ C )   =>    |- ( ph -> ( A i^i B ) C_ C )
Theorem	nelsn 37042	If an element doesn't match the item in an singleton it is not in the singleton (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A =/= B -> -. A e. { B } )
Theorem	zct 37043	The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ZZ ~<_ om
Theorem	iunxsngf2 37044*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- ( x = A -> B = C )   =>    |- ( A e. V -> U_ x e. { A } B = C )
Theorem	pwfin0 37045	A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ~P A i^i Fin ) =/= (/)
Theorem	uzct 37046	An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- Z = ( ZZ>= ` N )   =>    |- Z ~<_ om
Theorem	iunxsnf 37047*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- A e. _V   &    |- ( x = A -> B = C )   =>    |- U_ x e. { A } B = C
Theorem	fiiuncl 37048*	If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- ( ( ph /\ x e. A ) -> B e. D )   &    |- ( ( ph /\ y e. D /\ z e. D ) -> ( y u. z ) e. D )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> U_ x e. A B e. D )
Theorem	iunp1 37049*	The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ k B   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( k = ( N + 1 ) -> A = B )   =>    |- ( ph -> U_ k e. ( M ... ( N + 1 ) ) A = ( U_ k e. ( M ... N ) A u. B ) )
Theorem	fiunicl 37050*	If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A /\ y e. A ) -> ( x u. y ) e. A )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> U. A e. A )
21.30.2  Functions
Theorem	unima 37051	Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F Fn A /\ B C_ A /\ C C_ A ) -> ( F " ( B u. C ) ) = ( ( F " B ) u. ( F " C ) ) )
Theorem	feq1dd 37052	Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F = G )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> G : A --> B )
Theorem	fnresdmss 37053	A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F Fn A /\ A C_ B ) -> ( F |` B ) = F )
Theorem	fmptsnxp 37054*	Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. V /\ B e. W ) -> ( x e. { A } |-> B ) = ( { A } X. { B } ) )
Theorem	mptex2 37055*	If a class given as a map-to notation is a set, it's image values are set (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> ( t e. A |-> B ) : A --> C )   =>    |- ( ( ph /\ t e. A ) -> B e. C )
Theorem	fvmpt2bd 37056*	Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ( ph /\ x e. A /\ B e. C ) -> ( F ` x ) = B )
Theorem	rnmptfi 37057*	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- A = ( x e. B |-> C )   =>    |- ( B e. Fin -> ran A e. Fin )
Theorem	fresin2 37058	Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( F : A --> B -> ( F |` ( C i^i A ) ) = ( F |` C ) )
Theorem	rnmptc 37059*	Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. C )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> ran F = { B } )
Theorem	ffi 37060	A function with finite domain is finite (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F : A --> B /\ A e. Fin ) -> F e. Fin )
Theorem	suprnmpt 37061*	An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A =/= (/) )   &    |- ( ( ph /\ x e. A ) -> B e. RR )   &    |- ( ph -> E. y e. RR A. x e. A B <_ y )   &    |- F = ( x e. A |-> B )   &    |- C = sup ( ran F , RR , < )   =>    |- ( ph -> ( C e. RR /\ A. x e. A B <_ C ) )
Theorem	rnffi 37062	The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( F : A --> B /\ A e. Fin ) -> ran F e. Fin )
Theorem	mptelpm 37063*	A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
|- ( ( ph /\ x e. A ) -> B e. C )   &    |- ( ph -> A C_ D )   &    |- ( ph -> C e. V )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( x e. A |-> B ) e. ( C ^pm D ) )
Theorem	f1oeq3d 37064	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : C -1-1-onto-> A <-> F : C -1-1-onto-> B ) )
Theorem	fimass 37065	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A --> B -> ( F " X ) C_ B )
Theorem	rnmptpr 37066*	Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- F = ( x e. { A , B } |-> C )   &    |- ( x = A -> C = D )   &    |- ( x = B -> C = E )   =>    |- ( ph -> ran F = { D , E } )
Theorem	ffnd 37067	A mapping is a function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F : A --> B )   =>    |- ( ph -> F Fn A )
Theorem	resmpti 37068*	Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- B C_ A   =>    |- ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C )
Theorem	founiiun 37069*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A -onto-> B -> U. B = U_ x e. A ( F ` x ) )
Theorem	f1oeq2d 37070	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A = B )   =>    |- ( ph -> ( F : A -1-1-onto-> C <-> F : B -1-1-onto-> C ) )
Theorem	rnresun 37071	Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ran ( F |` ( A u. B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) )
Theorem	f1oeq1d 37072	Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F = G )   =>    |- ( ph -> ( F : A -1-1-onto-> B <-> G : A -1-1-onto-> B ) )
Theorem	dffo3f 37073*	An onto mapping expressed in terms of function values. As dffo3 6052 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x F   =>    |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
Theorem	rnresss 37074	The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ran ( A |` B ) C_ ran A
Theorem	mpteq1i 37075*	An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- A = B   =>    |- ( x e. A |-> C ) = ( x e. B |-> C )
Theorem	elrnmptd 37076*	The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> B )   &    |- ( ph -> E. x e. A C = B )   &    |- ( ph -> C e. V )   =>    |- ( ph -> C e. ran F )
Theorem	elrnmptf 37077	The range of a function in maps-to notation. Same as elrnmpt 5101, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- F = ( x e. A |-> B )   =>    |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
Theorem	mptss 37078*	Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )
Theorem	rnmptssrn 37079*	Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ( ph /\ x e. A ) -> E. y e. C B = D )   =>    |- ( ph -> ran ( x e. A |-> B ) C_ ran ( y e. C |-> D ) )
Theorem	disjf1 37080*	A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ( ph /\ x e. A ) -> B =/= (/) )   &    |- ( ph -> Disj x e. A B )   =>    |- ( ph -> F : A -1-1-> V )
Theorem	rnsnf 37081	The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> A e. V )   &    |- ( ph -> F : { A } --> B )   =>    |- ( ph -> ran F = { ( F ` A ) } )
Theorem	wessf1ornlem 37082*	Given a function F on a well ordered domain A there exists a subset of A such that F restricted to such subset is injective and onto the range of F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F Fn A )   &    |- ( ph -> A e. V )   &    |- ( ph -> R We A )   &    |- G = ( y e. ran F |-> ( iota_ x e. ( `' F " { y } ) A. z e. ( `' F " { y } ) -. z R x ) )   =>    |- ( ph -> E. x e. ~P A ( F |` x ) : x -1-1-onto-> ran F )
Theorem	wessf1orn 37083*	Given a function F on a well ordered domain A there exists a subset of A such that F restricted to such subset is injective and onto the range of F (without using the axiom of choice) (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ph -> F Fn A )   &    |- ( ph -> A e. V )   &    |- ( ph -> R We A )   =>    |- ( ph -> E. x e. ~P A ( F |` x ) : x -1-1-onto-> ran F )
Theorem	foelrnf 37084*	Property of a surjective function. As foelrn 6056 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x F   =>    |- ( ( F : A -onto-> B /\ C e. B ) -> E. x e. A C = ( F ` x ) )
Theorem	nelrnres 37085	If A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( -. A e. ran B -> -. A e. ran ( B |` C ) )
Theorem	disjrnmpt2 37086*	Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> B )   =>    |- (Disj x e. A B -> Disj y e. ran F y )
Theorem	elrnmpt1sf 37087*	Elementhood in an image set. Same as elrnmpt1s 5102, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/_ x C   &    |- F = ( x e. A |-> B )   &    |- ( x = D -> B = C )   =>    |- ( ( D e. A /\ C e. V ) -> C e. ran F )
Theorem	founiiun0 37088*	Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( F : A -onto-> ( B u. { (/) } ) -> U. B = U_ x e. A ( F ` x ) )
Theorem	disjf1o 37089*	A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F/ x ph   &    |- F = ( x e. A |-> B )   &    |- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> Disj x e. A B )   &    |- C = { x e. A | B =/= (/) }   &    |- D = ( ran F \ { (/) } )   =>    |- ( ph -> ( F |` C ) : C -1-1-onto-> D )
Theorem	fompt 37090*	Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- F = ( x e. A |-> C )   =>    |- ( F : A -onto-> B <-> ( A. x e. A C e. B /\ A. y e. B E. x e. A y = C ) )
Theorem	disjinfi 37091*	Only a finite number of disjoint sets can have a non empty intersection with a finite set C (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|- ( ( ph /\ x e. A ) -> B e. V )   &    |- ( ph -> Disj x e. A B )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> { x e. A | ( B i^i C ) =/= (/) } e. Fin )
21.30.3  Ordering on real numbers - Real and complex numbers basic operations
Theorem	sub2times 37092	Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. CC -> ( A - ( 2 x. A ) ) = -u A )
Theorem	xrltled 37093	'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> A <_ B )
Theorem	abssubrp 37094	The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( A e. CC /\ B e. CC /\ A =/= B ) -> ( abs ` ( A - B ) ) e. RR+ )
Theorem	elfzfzo 37095	Relationship between membership in a half open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. ( M..^ N ) <-> ( A e. ( M ... N ) /\ A < N ) )
Theorem	oddfl 37096	Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ( K e. ZZ /\ ( K mod 2 ) =/= 0 ) -> K = ( ( 2 x. ( |_ ` ( K / 2 ) ) ) + 1 ) )
Theorem	abscosbd 37097	Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( A e. RR -> ( abs ` ( cos ` A ) ) <_ 1 )
Theorem	mul13d 37098	Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A x. ( B x. C ) ) = ( C x. ( B x. A ) ) )
Theorem	negpilt0 37099	Negative pi is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- -u pi < 0
Theorem	dstregt0 37100*	A complex number A that is not real, has a distance from the reals that is strictly larger than 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|- ( ph -> A e. ( CC \ RR ) )   =>    |- ( ph -> E. x e. RR+ A. y e. RR x < ( abs ` ( A - y ) ) )