P. 52 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5101-5200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dmmptg 5101*	The domain of the mapping operation is the stated domain, if the function value is always a set. (Contributed by Mario Carneiro, 9-Feb-2013.) (Revised by Mario Carneiro, 14-Sep-2013.)
|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A )
Theorem	relco 5102	A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.)
|- Rel ( A o. B )
Theorem	dfco2 5103*	Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)
|- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
Theorem	dfco2a 5104*	Generalization of dfco2 5103, where C can have any value between dom A i^i ran B and _V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( dom A i^i ran B ) C_ C -> ( A o. B ) = U_ x e. C ( ( `' B " { x } ) X. ( A " { x } ) ) )
Theorem	coundi 5105	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( A o. ( B u. C ) ) = ( ( A o. B ) u. ( A o. C ) )
Theorem	coundir 5106	Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ( A u. B ) o. C ) = ( ( A o. C ) u. ( B o. C ) )
Theorem	cores 5107	Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
|- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )
Theorem	resco 5108	Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )
Theorem	imaco 5109	Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.)
|- ( ( A o. B ) " C ) = ( A " ( B " C ) )
Theorem	rnco 5110	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
|- ran ( A o. B ) = ran ( A |` ran B )
Theorem	rnco2 5111	The range of the composition of two classes. (Contributed by NM, 27-Mar-2008.)
|- ran ( A o. B ) = ( A " ran B )
Theorem	dmco 5112	The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
|- dom ( A o. B ) = ( `' B " dom A )
Theorem	coiun 5113*	Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
|- ( A o. U_ x e. C B ) = U_ x e. C ( A o. B )
Theorem	cocnvcnv1 5114	A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
|- ( `' `' A o. B ) = ( A o. B )
Theorem	cocnvcnv2 5115	A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
|- ( A o. `' `' B ) = ( A o. B )
Theorem	cores2 5116	Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
|- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )
Theorem	co02 5117	Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
|- ( A o. (/) ) = (/)
Theorem	co01 5118	Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
|- ( (/) o. A ) = (/)
Theorem	coi1 5119	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
|- ( Rel A -> ( A o. _I ) = A )
Theorem	coi2 5120	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
|- ( Rel A -> ( _I o. A ) = A )
Theorem	coires1 5121	Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- ( A o. ( _I |` B ) ) = ( A |` B )
Theorem	coass 5122	Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
|- ( ( A o. B ) o. C ) = ( A o. ( B o. C ) )
Theorem	relcnvtr 5123	A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
|- ( Rel R -> ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) )
Theorem	relssdmrn 5124	A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
|- ( Rel A -> A C_ ( dom A X. ran A ) )
Theorem	cnvssrndm 5125	The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
|- `' A C_ ( ran A X. dom A )
Theorem	cossxp 5126	Composition as a subset of the cross product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- ( A o. B ) C_ ( dom B X. ran A )
Theorem	cossxp2 5127	The composition of two relations is a relation, with bounds on its domain and codomain. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> R C_ ( A X. B ) )   &    |- ( ph -> S C_ ( B X. C ) )   =>    |- ( ph -> ( S o. R ) C_ ( A X. C ) )
Theorem	cocnvres 5128	Restricting a relation and a converse relation when they are composed together. (Contributed by BJ, 10-Jul-2022.)
|- ( S o. `' R ) = ( ( S |` dom R ) o. `' ( R |` dom S ) )
Theorem	cocnvss 5129	Upper bound for the composed of a relation and an inverse relation. (Contributed by BJ, 10-Jul-2022.)
|- ( S o. `' R ) C_ ( ran ( R |` dom S ) X. ran ( S |` dom R ) )
Theorem	relrelss 5130	Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
|- ( ( Rel A /\ Rel dom A ) <-> A C_ ( ( _V X. _V ) X. _V ) )
Theorem	unielrel 5131	The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
|- ( ( Rel R /\ A e. R ) -> U. A e. U. R )
Theorem	relfld 5132	The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
|- ( Rel R -> U. U. R = ( dom R u. ran R ) )
Theorem	relresfld 5133	Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
|- ( Rel R -> ( R |` U. U. R ) = R )
Theorem	relcoi2 5134	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
|- ( Rel R -> ( ( _I |` U. U. R ) o. R ) = R )
Theorem	relcoi1 5135	Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)
|- ( Rel R -> ( R o. ( _I |` U. U. R ) ) = R )
Theorem	unidmrn 5136	The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
|- U. U. `' A = ( dom A u. ran A )
Theorem	relcnvfld 5137	if R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
|- ( Rel R -> U. U. R = U. U. `' R )
Theorem	dfdm2 5138	Alternate definition of domain df-dm 4614 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
|- dom A = U. U. ( `' A o. A )
Theorem	unixpm 5139*	The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( E. x x e. ( A X. B ) -> U. U. ( A X. B ) = ( A u. B ) )
Theorem	unixp0im 5140	The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( ( A X. B ) = (/) -> U. ( A X. B ) = (/) )
Theorem	cnvexg 5141	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 17-Mar-1998.)
|- ( A e. V -> `' A e. _V )
Theorem	cnvex 5142	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.)
|- A e. _V   =>    |- `' A e. _V
Theorem	relcnvexb 5143	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
|- ( Rel R -> ( R e. _V <-> `' R e. _V ) )
Theorem	ressn 5144	Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- ( A |` { B } ) = ( { B } X. ( A " { B } ) )
Theorem	cnviinm 5145*	The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)
|- ( E. y y e. A -> `' |^|_ x e. A B = |^|_ x e. A `' B )
Theorem	cnvpom 5146*	The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
|- ( E. x x e. A -> ( R Po A <-> `' R Po A ) )
Theorem	cnvsom 5147*	The converse of a strict order relation is a strict order relation. (Contributed by Jim Kingdon, 19-Dec-2018.)
|- ( E. x x e. A -> ( R Or A <-> `' R Or A ) )
Theorem	coexg 5148	The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
|- ( ( A e. V /\ B e. W ) -> ( A o. B ) e. _V )
Theorem	coex 5149	The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
|- A e. _V   &    |- B e. _V   =>    |- ( A o. B ) e. _V
Theorem	xpcom 5150*	Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
|- ( E. x x e. B -> ( ( B X. C ) o. ( A X. B ) ) = ( A X. C ) )
2.6.7  Definite description binder (inverted iota)
Syntax	cio 5151	Extend class notation with Russell's definition description binder (inverted iota).
class ( iota x ph )
Theorem	iotajust 5152*	Soundness justification theorem for df-iota 5153. (Contributed by Andrew Salmon, 29-Jun-2011.)
|- U. { y | { x | ph } = { y } } = U. { z | { x | ph } = { z } }
Definition	df-iota 5153*	Define Russell's definition description binder, which can be read as "the unique x such that ph," where ph ordinarily contains x as a free variable. Our definition is meaningful only when there is exactly one x such that ph is true (see iotaval 5164); otherwise, it evaluates to the empty set (see iotanul 5168). Russell used the inverted iota symbol iota to represent the binder. Sometimes proofs need to expand an iota-based definition. That is, given "X = the x for which ... x ... x ..." holds, the proof needs to get to "... X ... X ...". A general strategy to do this is to use iotacl 5176 (for unbounded iota). This can be easier than applying a version that applies an explicit substitution, because substituting an iota into its own property always has a bound variable clash which must be first renamed or else guarded with NF. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( iota x ph ) = U. { y | { x | ph } = { y } }
Theorem	dfiota2 5154*	Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
Theorem	nfiota1 5155	Bound-variable hypothesis builder for the iota class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
|- F/_ x ( iota x ph )
Theorem	nfiotadw 5156*	Bound-variable hypothesis builder for the iota class. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x ( iota y ps ) )
Theorem	nfiotaw 5157*	Bound-variable hypothesis builder for the iota class. (Contributed by NM, 23-Aug-2011.)
|- F/ x ph   =>    |- F/_ x ( iota y ph )
Theorem	cbviota 5158	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( x = y -> ( ph <-> ps ) )   &    |- F/ y ph   &    |- F/ x ps   =>    |- ( iota x ph ) = ( iota y ps )
Theorem	cbviotav 5159*	Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota x ph ) = ( iota y ps )
Theorem	sb8iota 5160	Variable substitution in description binder. Compare sb8eu 2027. (Contributed by NM, 18-Mar-2013.)
|- F/ y ph   =>    |- ( iota x ph ) = ( iota y [ y / x ] ph )
Theorem	iotaeq 5161	Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )
Theorem	iotabi 5162	Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
|- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )
Theorem	uniabio 5163*	Part of Theorem 8.17 in [Quine] p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> U. { x | ph } = y )
Theorem	iotaval 5164*	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )
Theorem	iotauni 5165	Equivalence between two different forms of iota. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> ( iota x ph ) = U. { x | ph } )
Theorem	iotaint 5166	Equivalence between two different forms of iota. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( E! x ph -> ( iota x ph ) = |^| { x | ph } )
Theorem	iota1 5167	Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> ( ph <-> ( iota x ph ) = x ) )
Theorem	iotanul 5168	Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one x that satisfies ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( -. E! x ph -> ( iota x ph ) = (/) )
Theorem	euiotaex 5169	Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the iota class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)
|- ( E! x ph -> ( iota x ph ) e. _V )
Theorem	iotass 5170*	Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
|- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )
Theorem	iota4 5171	Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ph -> [. ( iota x ph ) / x ]. ph )
Theorem	iota4an 5172	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- ( E! x ( ph /\ ps ) -> [. ( iota x ( ph /\ ps ) ) / x ]. ph )
Theorem	iota5 5173*	A method for computing iota. (Contributed by NM, 17-Sep-2013.)
|- ( ( ph /\ A e. V ) -> ( ps <-> x = A ) )   =>    |- ( ( ph /\ A e. V ) -> ( iota x ps ) = A )
Theorem	iotabidv 5174*	Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( iota x ps ) = ( iota x ch ) )
Theorem	iotabii 5175	Formula-building deduction for iota. (Contributed by Mario Carneiro, 2-Oct-2015.)
|- ( ph <-> ps )   =>    |- ( iota x ph ) = ( iota x ps )
Theorem	iotacl 5176	Membership law for descriptions. This can useful for expanding an unbounded iota-based definition (see df-iota 5153). (Contributed by Andrew Salmon, 1-Aug-2011.)
|- ( E! x ph -> ( iota x ph ) e. { x | ph } )
Theorem	iota2df 5177	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   &    |- F/ x ph   &    |- ( ph -> F/ x ch )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	iota2d 5178*	A condition that allows us to represent "the unique element such that ph " with a class expression A. (Contributed by NM, 30-Dec-2014.)
|- ( ph -> B e. V )   &    |- ( ph -> E! x ps )   &    |- ( ( ph /\ x = B ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch <-> ( iota x ps ) = B ) )
Theorem	iota2 5179*	The unique element such that ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. B /\ E! x ph ) -> ( ps <-> ( iota x ph ) = A ) )
Theorem	sniota 5180	A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.)
|- ( E! x ph -> { x | ph } = { ( iota x ph ) } )
Theorem	csbiotag 5181*	Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
|- ( A e. V -> [_ A / x ]_ ( iota y ph ) = ( iota y [. A / x ]. ph ) )
2.6.8  Functions
Syntax	wfun 5182	Extend the definition of a wff to include the function predicate. (Read: A is a function.)
wff Fun A
Syntax	wfn 5183	Extend the definition of a wff to include the function predicate with a domain. (Read: A is a function on B.)
wff A Fn B
Syntax	wf 5184	Extend the definition of a wff to include the function predicate with domain and codomain. (Read: F maps A into B.)
wff F : A --> B
Syntax	wf1 5185	Extend the definition of a wff to include one-to-one functions. (Read: F maps A one-to-one into B.) The notation ("1-1" above the arrow) is from Definition 6.15(5) of [TakeutiZaring] p. 27.
wff F : A -1-1-> B
Syntax	wfo 5186	Extend the definition of a wff to include onto functions. (Read: F maps A onto B.) The notation ("onto" below the arrow) is from Definition 6.15(4) of [TakeutiZaring] p. 27.
wff F : A -onto-> B
Syntax	wf1o 5187	Extend the definition of a wff to include one-to-one onto functions. (Read: F maps A one-to-one onto B.) The notation ("1-1" above the arrow and "onto" below the arrow) is from Definition 6.15(6) of [TakeutiZaring] p. 27.
wff F : A -1-1-onto-> B
Syntax	cfv 5188	Extend the definition of a class to include the value of a function. (Read: The value of F at A, or " F of A.")
class ( F ` A )
Syntax	wiso 5189	Extend the definition of a wff to include the isomorphism property. (Read: H is an R, S isomorphism of A onto B.)
wff H Isom R , S ( A , B )
Definition	df-fun 5190	Define predicate that determines if some class A is a function. Definition 10.1 of [Quine] p. 65. For example, the expression Fun _I is true (funi 5220). This is not the same as defining a specific function's mapping, which is typically done using the format of cmpt 4043 with the maps-to notation (see df-mpt 4045). Contrast this predicate with the predicates to determine if some class is a function with a given domain (df-fn 5191), a function with a given domain and codomain (df-f 5192), a one-to-one function (df-f1 5193), an onto function (df-fo 5194), or a one-to-one onto function (df-f1o 5195). For alternate definitions, see dffun2 5198, dffun4 5199, dffun6 5202, dffun7 5215, dffun8 5216, and dffun9 5217. (Contributed by NM, 1-Aug-1994.)
|- ( Fun A <-> ( Rel A /\ ( A o. `' A ) C_ _I ) )
Definition	df-fn 5191	Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
|- ( A Fn B <-> ( Fun A /\ dom A = B ) )
Definition	df-f 5192	Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27. (Contributed by NM, 1-Aug-1994.)
|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
Definition	df-f1 5193	Define a one-to-one function. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
Definition	df-fo 5194	Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
Definition	df-f1o 5195	Define a one-to-one onto function. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow). (Contributed by NM, 1-Aug-1994.)
|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
Definition	df-fv 5196*	Define the value of a function, ( F ` A ), also known as function application. For example, ( _I ` (/) ) = (/). Typically, function F is defined using maps-to notation (see df-mpt 4045), but this is not required. For example, F = { <. 2 , 6 >. , <. 3 , 9 >. } -> ( F ` 3 ) = 9. We will later define two-argument functions using ordered pairs as ( A F B ) = ( F ` <. A , B >. ). This particular definition is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful. The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar F ( A ) notation for a function's value at A, i.e., " F of A," but without context-dependent notational ambiguity. (Contributed by NM, 1-Aug-1994.) Revised to use iota. (Revised by Scott Fenton, 6-Oct-2017.)
|- ( F ` A ) = ( iota x A F x )
Definition	df-isom 5197*	Define the isomorphism predicate. We read this as " H is an R, S isomorphism of A onto B". Normally, R and S are ordering relations on A and B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that R and S are subscripts. (Contributed by NM, 4-Mar-1997.)
|- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
Theorem	dffun2 5198*	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.)
|- ( Fun A <-> ( Rel A /\ A. x A. y A. z ( ( x A y /\ x A z ) -> y = z ) ) )
Theorem	dffun4 5199*	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
|- ( Fun A <-> ( Rel A /\ A. x A. y A. z ( ( <. x , y >. e. A /\ <. x , z >. e. A ) -> y = z ) ) )
Theorem	dffun5r 5200*	A way of proving a relation is a function, analogous to mo2r 2066. (Contributed by Jim Kingdon, 27-May-2020.)
|- ( ( Rel A /\ A. x E. z A. y ( <. x , y >. e. A -> y = z ) ) -> Fun A )