P. 66 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6501-6600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	suppssrst 6501*	A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssrgst 6502*	A function is zero outside its support. Version of suppssrst 6501 avoiding ax-coll 4246 by assuming F is a set rather than its domain A. (Contributed by SN, 5-May-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> F e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
Theorem	suppssfvg 6503*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
|- ( ph -> ( ( x e. D |-> A ) supp Y ) C_ L )   &    |- ( ph -> ( F ` Y ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   &    |- ( ph -> Y e. U )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( ( x e. D |-> ( F ` A ) ) supp Z ) C_ L )
Theorem	suppofss1dcl 6504*	Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( Z X x ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( F supp Z ) )
Theorem	suppofss2dcl 6505*	Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
|- ( ph -> A e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> G : A --> B )   &    |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u X v ) e. B )   &    |- ( ( ph /\ x e. B ) -> ( x X Z ) = Z )   =>    |- ( ph -> ( ( F oF X G ) supp Z ) C_ ( G supp Z ) )
Theorem	suppcofn 6506	The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) (Revised by SN, 15-Sep-2023.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F o. G ) supp Z ) = ( `' G " ( F supp Z ) ) )
Theorem	supp0cosupp0fn 6507	The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( F supp Z ) = (/) -> ( ( F o. G ) supp Z ) = (/) ) )
Theorem	imacosuppfn 6508	The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
|- ( ( ( F e. V /\ G e. W ) /\ ( Fun F /\ Fun G ) ) -> ( ( Fun G /\ ( F supp Z ) C_ ran G ) -> ( G " ( ( F o. G ) supp Z ) ) = ( F supp Z ) ) )
2.6.17  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 6090) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 6166, ovmpox 6217 and fmpox 6436). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 6509*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4920. (Contributed by AV, 25-Oct-2020.)
|- F/_ x E   &    |- ( x = C -> B = E )   =>    |- ( <. C , D >. e. U_ x e. A ( { x } X. B ) <-> ( C e. A /\ D e. E ) )
Theorem	mpoxopn0yelv 6510*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> C )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) -> K e. V ) )
Theorem	mpoxopoveq 6511*	Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
Theorem	mpoxopovel 6512*	Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.)
|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )   =>    |- ( ( V e. X /\ W e. Y ) -> ( N e. ( <. V , W >. F K ) <-> ( K e. V /\ N e. V /\ [. <. V , W >. / x ]. [. K / y ]. [. N / n ]. ph ) ) )
Theorem	rbropapd 6513*	Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) ) )
Theorem	rbropap 6514*	Properties of a pair in a restricted binary relation M expressed as an ordered-pair class abstraction: M is the binary relation W restricted by the condition ps. (Contributed by AV, 31-Jan-2021.)
|- ( ph -> M = { <. f , p >. | ( f W p /\ ps ) } )   &    |- ( ( f = F /\ p = P ) -> ( ps <-> ch ) )   =>    |- ( ( ph /\ F e. X /\ P e. Y ) -> ( F M P <-> ( F W P /\ ch ) ) )
2.6.18  Function transposition
Syntax	ctpos 6515	The transposition of a function.
class tpos F
Definition	df-tpos 6516*	Define the transposition of a function, which is a function G = tpos F satisfying G ( x , y ) = F ( y , x ). (Contributed by Mario Carneiro, 10-Sep-2015.)
|- tpos F = ( F o. ( x e. ( `' dom F u. { (/) } ) |-> U. `' { x } ) )
Theorem	tposss 6517	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F C_ G -> tpos F C_ tpos G )
Theorem	tposeq 6518	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F = G -> tpos F = tpos G )
Theorem	tposeqd 6519	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> F = G )   =>    |- ( ph -> tpos F = tpos G )
Theorem	tposssxp 6520	The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.)
|- tpos F C_ ( ( `' dom F u. { (/) } ) X. ran F )
Theorem	reltpos 6521	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- Rel tpos F
Theorem	brtpos2 6522	Value of the transposition at a pair <. A , B >.. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( B e. V -> ( Atpos F B <-> ( A e. ( `' dom F u. { (/) } ) /\ U. `' { A } F B ) ) )
Theorem	brtpos0 6523	The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( A e. V -> ( (/)tpos F A <-> (/) F A ) )
Theorem	reldmtpos 6524	Necessary and sufficient condition for dom tpos F to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom tpos F <-> -. (/) e. dom F )
Theorem	brtposg 6525	The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B >.tpos F C <-> <. B , A >. F C ) )
Theorem	ottposg 6526	The transposition swaps the first two elements in a collection of ordered triples. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. A , B , C >. e. tpos F <-> <. B , A , C >. e. F ) )
Theorem	dmtpos 6527	The domain of tpos F when dom F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> dom tpos F = `' dom F )
Theorem	rntpos 6528	The range of tpos F when dom F is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> ran tpos F = ran F )
Theorem	tposexg 6529	The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F e. V -> tpos F e. _V )
Theorem	ovtposg 6530	The transposition swaps the arguments in a two-argument function. When F is a matrix, which is to say a function from ( 1 ... m ) X. ( 1 ... n ) to the reals or some ring, tpos F is the transposition of F, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( ( A e. V /\ B e. W ) -> ( Atpos F B ) = ( B F A ) )
Theorem	tposfun 6531	The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Fun F -> Fun tpos F )
Theorem	dftpos2 6532*	Alternate definition of tpos when F has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> tpos F = ( F o. ( x e. `' dom F |-> U. `' { x } ) ) )
Theorem	dftpos3 6533*	Alternate definition of tpos when F has relational domain. Compare df-cnv 4782. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> tpos F = { <. <. x , y >. , z >. | <. y , x >. F z } )
Theorem	dftpos4 6534*	Alternate definition of tpos. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- tpos F = ( F o. ( x e. ( ( _V X. _V ) u. { (/) } ) |-> U. `' { x } ) )
Theorem	tpostpos 6535	Value of the double transposition for a general class F. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- tpos tpos F = ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) )
Theorem	tpostpos2 6536	Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
|- ( ( Rel F /\ Rel dom F ) -> tpos tpos F = F )
Theorem	tposfn2 6537	The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
Theorem	tposfo2 6538	Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )
Theorem	tposf2 6539	The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )
Theorem	tposf12 6540	Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -1-1-> B -> tpos F : `' A -1-1-> B ) )
Theorem	tposf1o2 6541	Condition of a bijective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -1-1-onto-> B -> tpos F : `' A -1-1-onto-> B ) )
Theorem	tposfo 6542	The domain and codomain/range of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( F : ( A X. B ) -onto-> C -> tpos F : ( B X. A ) -onto-> C )
Theorem	tposf 6543	The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( F : ( A X. B ) --> C -> tpos F : ( B X. A ) --> C )
Theorem	tposfn 6544	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( F Fn ( A X. B ) -> tpos F Fn ( B X. A ) )
Theorem	tpos0 6545	Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
|- tpos (/) = (/)
Theorem	tposco 6546	Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- tpos ( F o. G ) = ( F o. tpos G )
Theorem	tpossym 6547*	Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( F Fn ( A X. A ) -> (tpos F = F <-> A. x e. A A. y e. A ( x F y ) = ( y F x ) ) )
Theorem	tposeqi 6548	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = G   =>    |- tpos F = tpos G
Theorem	tposex 6549	A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F e. _V   =>    |- tpos F e. _V
Theorem	nftpos 6550	Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F/_ x F   =>    |- F/_ xtpos F
Theorem	tposoprab 6551*	Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = { <. <. x , y >. , z >. | ph }   =>    |- tpos F = { <. <. y , x >. , z >. | ph }
Theorem	tposmpo 6552*	Transposition of a two-argument mapping. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- tpos F = ( y e. B , x e. A |-> C )
2.6.19  Undefined values
Theorem	pwuninel2 6553	The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( U. A e. V -> -. ~P U. A e. A )
Theorem	2pwuninelg 6554	The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.)
|- ( A e. V -> -. ~P ~P U. A e. A )
2.6.20  Functions on ordinals; strictly monotone ordinal functions
Theorem	iunon 6555*	The indexed union of a set of ordinal numbers B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
|- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )
Syntax	wsmo 6556	Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.
wff Smo A
Definition	df-smo 6557*	Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
|- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )
Theorem	dfsmo2 6558*	Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)
|- ( Smo F <-> ( F : dom F --> On /\ Ord dom F /\ A. x e. dom F A. y e. x ( F ` y ) e. ( F ` x ) ) )
Theorem	issmo 6559*	Conditions for which A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
|- A : B --> On   &    |- Ord B   &    |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )   &    |- dom A = B   =>    |- Smo A
Theorem	issmo2 6560*	Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ( F : A --> B -> ( ( B C_ On /\ Ord A /\ A. x e. A A. y e. x ( F ` y ) e. ( F ` x ) ) -> Smo F ) )
Theorem	smoeq 6561	Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( A = B -> ( Smo A <-> Smo B ) )
Theorem	smodm 6562	The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( Smo A -> Ord dom A )
Theorem	smores 6563	A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
|- ( ( Smo A /\ B e. dom A ) -> Smo ( A |` B ) )
Theorem	smores3 6564	A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
|- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( A |` C ) )
Theorem	smores2 6565	A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
|- ( ( Smo F /\ Ord A ) -> Smo ( F |` A ) )
Theorem	smodm2 6566	The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ( ( F Fn A /\ Smo F ) -> Ord A )
Theorem	smofvon2dm 6567	The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ( ( Smo F /\ B e. dom F ) -> ( F ` B ) e. On )
Theorem	iordsmo 6568	The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- Ord A   =>    |- Smo ( _I |` A )
Theorem	smo0 6569	The empty set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- Smo (/)
Theorem	smofvon 6570	If B is a strictly monotone ordinal function, and A is in the domain of B, then the value of the function at A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- ( ( Smo B /\ A e. dom B ) -> ( B ` A ) e. On )
Theorem	smoel 6571	If x is less than y then a strictly monotone function's value will be strictly less at x than at y. (Contributed by Andrew Salmon, 22-Nov-2011.)
|- ( ( Smo B /\ A e. dom B /\ C e. A ) -> ( B ` C ) e. ( B ` A ) )
Theorem	smoiun 6572*	The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
|- ( ( Smo B /\ A e. dom B ) -> U_ x e. A ( B ` x ) C_ ( B ` A ) )
Theorem	smoiso 6573	If F is an isomorphism from an ordinal A onto B, which is a subset of the ordinals, then F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ B C_ On ) -> Smo F )
Theorem	smoel2 6574	A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )
2.6.21  "Strong" transfinite recursion
Syntax	crecs 6575	Notation for a function defined by strong transfinite recursion.
class recs ( F )
Definition	df-recs 6576*	Define a function recs ( F ) on On, the class of ordinal numbers, by transfinite recursion given a rule F which sets the next value given all values so far. See df-irdg 6641 for more details on why this definition is desirable. Unlike df-irdg 6641 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See tfri1d 6606 and tfri2d 6607 for the primary contract of this definition. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
Theorem	recseq 6577	Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- ( F = G -> recs ( F ) = recs ( G ) )
Theorem	nfrecs 6578	Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- F/_ x F   =>    |- F/_ xrecs ( F )
Theorem	tfrlem1 6579*	A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.)
|- ( ph -> A e. On )   &    |- ( ph -> ( Fun F /\ A C_ dom F ) )   &    |- ( ph -> ( Fun G /\ A C_ dom G ) )   &    |- ( ph -> A. x e. A ( F ` x ) = ( B ` ( F |` x ) ) )   &    |- ( ph -> A. x e. A ( G ` x ) = ( B ` ( G |` x ) ) )   =>    |- ( ph -> A. x e. A ( F ` x ) = ( G ` x ) )
Theorem	tfrlem3ag 6580*	Lemma for transfinite recursion. This lemma just changes some bound variables in A for later use. (Contributed by Jim Kingdon, 5-Jul-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )
Theorem	tfrlem3a 6581*	Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- G e. _V   =>    |- ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) )
Theorem	tfrlem3 6582*	Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
Theorem	tfrlem3-2d 6583*	Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)
|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   =>    |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )
Theorem	tfrlem4 6584*	Lemma for transfinite recursion. A is the class of all "acceptable" functions, and F is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- ( g e. A -> Fun g )
Theorem	tfrlem5 6585*	Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- ( ( g e. A /\ h e. A ) -> ( ( x g u /\ x h v ) -> u = v ) )
Theorem	recsfval 6586*	Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- recs ( F ) = U. A
Theorem	tfrlem6 6587*	Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- Rel recs ( F )
Theorem	tfrlem7 6588*	Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- Fun recs ( F )
Theorem	tfrlem8 6589*	Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- Ord dom recs ( F )
Theorem	tfrlem9 6590*	Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   =>    |- ( B e. dom recs ( F ) -> (recs ( F ) ` B ) = ( F ` (recs ( F ) |` B ) ) )
Theorem	tfrfun 6591	Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- Fun recs ( F )
Theorem	tfr2a 6592	A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- F = recs ( G )   =>    |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )
Theorem	tfr0dm 6593	Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)
|- F = recs ( G )   =>    |- ( ( G ` (/) ) e. V -> (/) e. dom F )
Theorem	tfr0 6594	Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
|- F = recs ( G )   =>    |- ( ( G ` (/) ) e. V -> ( F ` (/) ) = ( G ` (/) ) )
Theorem	tfrlemisucfn 6595*	We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 2-Jul-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- ( ph -> z e. On )   &    |- ( ph -> g Fn z )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) Fn suc z )
Theorem	tfrlemisucaccv 6596*	We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- ( ph -> z e. On )   &    |- ( ph -> g Fn z )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( F ` g ) >. } ) e. A )
Theorem	tfrlemibacc 6597*	Each element of B is an acceptable function. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }   &    |- ( ph -> x e. On )   &    |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )   =>    |- ( ph -> B C_ A )
Theorem	tfrlemibxssdm 6598*	The union of B is defined on all ordinals. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }   &    |- ( ph -> x e. On )   &    |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )   =>    |- ( ph -> x C_ dom U. B )
Theorem	tfrlemibfn 6599*	The union of B is a function defined on x. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }   &    |- ( ph -> x e. On )   &    |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )   =>    |- ( ph -> U. B Fn x )
Theorem	tfrlemibex 6600*	The set B exists. Lemma for tfrlemi1 6603. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }   &    |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )   &    |- B = { h | E. z e. x E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( F ` g ) >. } ) ) }   &    |- ( ph -> x e. On )   &    |- ( ph -> A. z e. x E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )   =>    |- ( ph -> B e. _V )