P. 65 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disjsnxp 6401*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|- Disj j e. A ( { j } X. B )
Theorem	elmpom 6402*	If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( D e. F -> E. z z e. A )
2.6.16  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 6022) 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 6098, ovmpox 6149 and fmpox 6364). 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 6403*	Membership in a union of Cartesian products, using bound-variable hypothesis for E instead of distinct variable conditions as in opeliunxp2 4870. (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 6404*	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 6405*	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 6406*	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 6407*	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 6408*	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 ) ) )
2.6.17  Function transposition
Syntax	ctpos 6409	The transposition of a function.
class tpos F
Definition	df-tpos 6410*	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 6411	Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F C_ G -> tpos F C_ tpos G )
Theorem	tposeq 6412	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F = G -> tpos F = tpos G )
Theorem	tposeqd 6413	Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)
|- ( ph -> F = G )   =>    |- ( ph -> tpos F = tpos G )
Theorem	tposssxp 6414	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 6415	The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- Rel tpos F
Theorem	brtpos2 6416	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 6417	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 6418	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 6419	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 6420	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 6421	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 6422	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 6423	The transposition of a set is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( F e. V -> tpos F e. _V )
Theorem	ovtposg 6424	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 6425	The transposition of a function is a function. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Fun F -> Fun tpos F )
Theorem	dftpos2 6426*	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 6427*	Alternate definition of tpos when F has relational domain. Compare df-cnv 4733. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- ( Rel dom F -> tpos F = { <. <. x , y >. , z >. | <. y , x >. F z } )
Theorem	dftpos4 6428*	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 6429	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 6430	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 6431	The domain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F Fn A -> tpos F Fn `' A ) )
Theorem	tposfo2 6432	Condition for a surjective transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A -onto-> B -> tpos F : `' A -onto-> B ) )
Theorem	tposf2 6433	The domain and codomain of a transposition. (Contributed by NM, 10-Sep-2015.)
|- ( Rel A -> ( F : A --> B -> tpos F : `' A --> B ) )
Theorem	tposf12 6434	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 6435	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 6436	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 6437	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 6438	Functionality of a transposition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- ( F Fn ( A X. B ) -> tpos F Fn ( B X. A ) )
Theorem	tpos0 6439	Transposition of the empty set. (Contributed by NM, 10-Sep-2015.)
|- tpos (/) = (/)
Theorem	tposco 6440	Transposition of a composition. (Contributed by Mario Carneiro, 4-Oct-2015.)
|- tpos ( F o. G ) = ( F o. tpos G )
Theorem	tpossym 6441*	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 6442	Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F = G   =>    |- tpos F = tpos G
Theorem	tposex 6443	A transposition is a set. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F e. _V   =>    |- tpos F e. _V
Theorem	nftpos 6444	Hypothesis builder for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.)
|- F/_ x F   =>    |- F/_ xtpos F
Theorem	tposoprab 6445*	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 6446*	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 )
2.6.18  Undefined values
Theorem	pwuninel2 6447	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 6448	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 )
2.6.19  Functions on ordinals; strictly monotone ordinal functions
Theorem	iunon 6449*	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 6450	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 6451*	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 6452*	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 6453*	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 6454*	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 6455	Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( A = B -> ( Smo A <-> Smo B ) )
Theorem	smodm 6456	The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ( Smo A -> Ord dom A )
Theorem	smores 6457	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 6458	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 6459	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 6460	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 6461	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 6462	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 6463	The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- Smo (/)
Theorem	smofvon 6464	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 6465	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 6466*	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 6467	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 6468	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 ) )
2.6.20  "Strong" transfinite recursion
Syntax	crecs 6469	Notation for a function defined by strong transfinite recursion.
class recs ( F )
Definition	df-recs 6470*	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 6535 for more details on why this definition is desirable. Unlike df-irdg 6535 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 6500 and tfri2d 6501 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 6471	Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- ( F = G -> recs ( F ) = recs ( G ) )
Theorem	nfrecs 6472	Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- F/_ x F   =>    |- F/_ xrecs ( F )
Theorem	tfrlem1 6473*	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 6474*	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 6475*	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 6476*	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 6477*	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 6478*	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 6479*	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 6480*	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 6481*	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 6482*	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 6483*	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 6484*	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 6485	Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- Fun recs ( F )
Theorem	tfr2a 6486	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 6487	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 6488	Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
|- F = recs ( G )   =>    |- ( ( G ` (/) ) e. V -> ( F ` (/) ) = ( G ` (/) ) )
Theorem	tfrlemisucfn 6489*	We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6497. (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 6490*	We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6497. (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 6491*	Each element of B is an acceptable function. Lemma for tfrlemi1 6497. (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 6492*	The union of B is defined on all ordinals. Lemma for tfrlemi1 6497. (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 6493*	The union of B is a function defined on x. Lemma for tfrlemi1 6497. (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 6494*	The set B exists. Lemma for tfrlemi1 6497. (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 )
Theorem	tfrlemiubacc 6495*	The union of B satisfies the recursion rule (lemma for tfrlemi1 6497). (Contributed by Jim Kingdon, 22-Apr-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 -> A. u e. x ( U. B ` u ) = ( F ` ( U. B |` u ) ) )
Theorem	tfrlemiex 6496*	Lemma for tfrlemi1 6497. (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 -> E. f ( f Fn x /\ A. u e. x ( f ` u ) = ( F ` ( f |` u ) ) ) )
Theorem	tfrlemi1 6497*	We can define an acceptable function on any ordinal. As with many of the transfinite recursion theorems, we have a hypothesis that states that F is a function and that it is defined for all ordinals. (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 /\ C e. On ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( F ` ( g |` u ) ) ) )
Theorem	tfrlemi14d 6498*	The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-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 -> dom recs ( F ) = On )
Theorem	tfrexlem 6499*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-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 /\ C e. V ) -> (recs ( F ) ` C ) e. _V )
Theorem	tfri1d 6500*	Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition. The condition is that G is defined "everywhere", which is stated here as ( G ` x ) e. _V. Alternately, A. x e. On A. f ( f Fn x -> f e. dom G ) would suffice. Given a function G satisfying that condition, we define a class A of all "acceptable" functions. The final function we're interested in is the union F = recs ( G ) of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
|- F = recs ( G )   &    |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )   =>    |- ( ph -> F Fn On )