P. 62 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	smo0 6101	The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- Smo (/)
Theorem	smofvon 6102	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 6103	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 6104*	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 6105	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 6106	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 ) )
2.6.19  "Strong" transfinite recursion
Syntax	crecs 6107	Notation for a function defined by strong transfinite recursion.
class recs ( F )
Definition	df-recs 6108*	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 6173 for more details on why this definition is desirable. Unlike df-irdg 6173 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 6138 and tfri2d 6139 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 6109	Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- ( F = G -> recs ( F ) = recs ( G ) )
Theorem	nfrecs 6110	Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
|- F/_ x F   =>    |- F/_ xrecs ( F )
Theorem	tfrlem1 6111*	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 6112*	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 6113*	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 6114*	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 6115*	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 6116*	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 6117*	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 6118*	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 6119*	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 6120*	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 6121*	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 6122*	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 6123	Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- Fun recs ( F )
Theorem	tfr2a 6124	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 6125	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 6126	Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.)
|- F = recs ( G )   =>    |- ( ( G ` (/) ) e. V -> ( F ` (/) ) = ( G ` (/) ) )
Theorem	tfrlemisucfn 6127*	We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6135. (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 6128*	We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6135. (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 6129*	Each element of B is an acceptable function. Lemma for tfrlemi1 6135. (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 6130*	The union of B is defined on all ordinals. Lemma for tfrlemi1 6135. (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 6131*	The union of B is a function defined on x. Lemma for tfrlemi1 6135. (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 6132*	The set B exists. Lemma for tfrlemi1 6135. (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 6133*	The union of B satisfies the recursion rule (lemma for tfrlemi1 6135). (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 6134*	Lemma for tfrlemi1 6135. (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 6135*	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 6136*	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 6137*	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 6138*	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 )
Theorem	tfri2d 6139*	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6168). Here we show that the function F has the property that for any function G satisfying that condition, the "next" value of F is G recursively applied to all "previous" values of F. (Contributed by Jim Kingdon, 4-May-2019.)
|- F = recs ( G )   &    |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )   =>    |- ( ( ph /\ A e. On ) -> ( F ` A ) = ( G ` ( F |` A ) ) )
Theorem	tfr1onlem3ag 6140*	Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6112 but for tfr1on 6153 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.)
|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   =>    |- ( H e. V -> ( H e. A <-> E. z e. X ( H Fn z /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) )
Theorem	tfr1onlem3 6141*	Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3 6114 but for tfr1on 6153 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)
|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   =>    |- A = { g | E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) }
Theorem	tfr1onlemssrecs 6142*	Lemma for tfr1on 6153. The union of functions acceptable for tfr1on 6153 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.)
|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> Ord X )   =>    |- ( ph -> U. A C_ recs ( G ) )
Theorem	tfr1onlemsucfn 6143*	We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6153. (Contributed by Jim Kingdon, 12-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> z e. X )   &    |- ( ph -> g Fn z )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) Fn suc z )
Theorem	tfr1onlemsucaccv 6144*	Lemma for tfr1on 6153. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> Y e. X )   &    |- ( ph -> z e. Y )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> g Fn z )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) e. A )
Theorem	tfr1onlembacc 6145*	Lemma for tfr1on 6153. Each element of B is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> B C_ A )
Theorem	tfr1onlembxssdm 6146*	Lemma for tfr1on 6153. The union of B is defined on all elements of X. (Contributed by Jim Kingdon, 14-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> D C_ dom U. B )
Theorem	tfr1onlembfn 6147*	Lemma for tfr1on 6153. The union of B is a function defined on x. (Contributed by Jim Kingdon, 15-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> U. B Fn D )
Theorem	tfr1onlembex 6148*	Lemma for tfr1on 6153. The set B exists. (Contributed by Jim Kingdon, 14-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> B e. _V )
Theorem	tfr1onlemubacc 6149*	Lemma for tfr1on 6153. The union of B satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )
Theorem	tfr1onlemex 6150*	Lemma for tfr1on 6153. (Contributed by Jim Kingdon, 16-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g Fn z /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> E. f ( f Fn D /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) )
Theorem	tfr1onlemaccex 6151*	We can define an acceptable function on any element of X. As with many of the transfinite recursion theorems, we have hypotheses that state that F is a function and that it is defined up to X. (Contributed by Jim Kingdon, 16-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   =>    |- ( ( ph /\ C e. X ) -> E. g ( g Fn C /\ A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) )
Theorem	tfr1onlemres 6152*	Lemma for tfr1on 6153. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. X )   =>    |- ( ph -> Y C_ dom F )
Theorem	tfr1on 6153*	Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f Fn x ) -> ( G ` f ) e. _V )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. X )   =>    |- ( ph -> Y C_ dom F )
Theorem	tfri1dALT 6154*	Alternate proof of tfri1d 6138 in terms of tfr1on 6153. Although this does show that the tfr1on 6153 proof is general enough to also prove tfri1d 6138, the tfri1d 6138 proof is simpler in places because it does not need to deal with X being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )   =>    |- ( ph -> F Fn On )
Theorem	tfrcllemssrecs 6155*	Lemma for tfrcl 6167. The union of functions acceptable for tfrcl 6167 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> Ord X )   =>    |- ( ph -> U. A C_ recs ( G ) )
Theorem	tfrcllemsucfn 6156*	We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6167. (Contributed by Jim Kingdon, 24-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> z e. X )   &    |- ( ph -> g : z --> S )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) : suc z --> S )
Theorem	tfrcllemsucaccv 6157*	Lemma for tfrcl 6167. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ph -> Y e. X )   &    |- ( ph -> z e. Y )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> g : z --> S )   &    |- ( ph -> g e. A )   =>    |- ( ph -> ( g u. { <. z , ( G ` g ) >. } ) e. A )
Theorem	tfrcllembacc 6158*	Lemma for tfrcl 6167. Each element of B is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> B C_ A )
Theorem	tfrcllembxssdm 6159*	Lemma for tfrcl 6167. The union of B is defined on all elements of X. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> D C_ dom U. B )
Theorem	tfrcllembfn 6160*	Lemma for tfrcl 6167. The union of B is a function defined on x. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> U. B : D --> S )
Theorem	tfrcllembex 6161*	Lemma for tfrcl 6167. The set B exists. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> B e. _V )
Theorem	tfrcllemubacc 6162*	Lemma for tfrcl 6167. The union of B satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> A. u e. D ( U. B ` u ) = ( G ` ( U. B |` u ) ) )
Theorem	tfrcllemex 6163*	Lemma for tfrcl 6167. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- B = { h | E. z e. D E. g ( g : z --> S /\ g e. A /\ h = ( g u. { <. z , ( G ` g ) >. } ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> D e. X )   &    |- ( ph -> A. z e. D E. g ( g : z --> S /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )   =>    |- ( ph -> E. f ( f : D --> S /\ A. u e. D ( f ` u ) = ( G ` ( f |` u ) ) ) )
Theorem	tfrcllemaccex 6164*	We can define an acceptable function on any element of X. As with many of the transfinite recursion theorems, we have hypotheses that state that F is a function and that it is defined up to X. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   =>    |- ( ( ph /\ C e. X ) -> E. g ( g : C --> S /\ A. u e. C ( g ` u ) = ( G ` ( g |` u ) ) ) )
Theorem	tfrcllemres 6165*	Lemma for tfr1on 6153. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- A = { f | E. x e. X ( f : x --> S /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. X )   =>    |- ( ph -> Y C_ dom F )
Theorem	tfrcldm 6166*	Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. U. X )   =>    |- ( ph -> Y e. dom F )
Theorem	tfrcl 6167*	Closure for transfinite recursion. As with tfr1on 6153, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)
|- F = recs ( G )   &    |- ( ph -> Fun G )   &    |- ( ph -> Ord X )   &    |- ( ( ph /\ x e. X /\ f : x --> S ) -> ( G ` f ) e. S )   &    |- ( ( ph /\ x e. U. X ) -> suc x e. X )   &    |- ( ph -> Y e. U. X )   =>    |- ( ph -> ( F ` Y ) e. S )
Theorem	tfri1 6168*	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 )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- F Fn On
Theorem	tfri2 6169*	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6168). Here we show that the function F has the property that for any function G satisfying that condition, the "next" value of F is G recursively applied to all "previous" values of F. (Contributed by Jim Kingdon, 4-May-2019.)
|- F = recs ( G )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- ( A e. On -> ( F ` A ) = ( G ` ( F |` A ) ) )
Theorem	tfri3 6170*	Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule G ( as described at tfri1 6168). Finally, we show that F is unique. We do this by showing that any class B with the same properties of F that we showed in parts 1 and 2 is identical to F. (Contributed by Jim Kingdon, 4-May-2019.)
|- F = recs ( G )   &    |- ( Fun G /\ ( G ` x ) e. _V )   =>    |- ( ( B Fn On /\ A. x e. On ( B ` x ) = ( G ` ( B |` x ) ) ) -> B = F )
Theorem	tfrex 6171*	The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- F = recs ( G )   &    |- ( ph -> A. x ( Fun G /\ ( G ` x ) e. _V ) )   =>    |- ( ( ph /\ A e. V ) -> ( F ` A ) e. _V )
2.6.20  Recursive definition generator
Syntax	crdg 6172	Extend class notation with the recursive definition generator, with characteristic function F and initial value I.
class rec ( F , I )
Definition	df-irdg 6173*	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function F and initial value I. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation (especially when df-recs 6108 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple. In classical logic it would be easier to divide this definition into cases based on whether the domain of g is zero, a successor, or a limit ordinal. Cases do not (in general) work that way in intuitionistic logic, so instead we choose a definition which takes the union of all the results of the characteristic function for ordinals in the domain of g. This means that this definition has the expected properties for increasing and continuous ordinal functions, which include ordinal addition and multiplication. For finite recursion we also define df-frec 6194 and for suitable characteristic functions df-frec 6194 yields the same result as rec restricted to om, as seen at frecrdg 6211. Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Jim Kingdon, 19-May-2019.)
|- rec ( F , I ) = recs ( ( g e. _V |-> ( I u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) )
Theorem	rdgeq1 6174	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
|- ( F = G -> rec ( F , A ) = rec ( G , A ) )
Theorem	rdgeq2 6175	Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)
|- ( A = B -> rec ( F , A ) = rec ( F , B ) )
Theorem	rdgfun 6176	The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
|- Fun rec ( F , A )
Theorem	rdgtfr 6177*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.)
|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgruledefgg 6178*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- ( ( F Fn _V /\ A e. V ) -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgruledefg 6179*	The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- F Fn _V   =>    |- ( A e. V -> ( Fun ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) /\ ( ( g e. _V |-> ( A u. U_ x e. dom g ( F ` ( g ` x ) ) ) ) ` f ) e. _V ) )
Theorem	rdgexggg 6180	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- ( ( F Fn _V /\ A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V )
Theorem	rdgexgg 6181	The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.)
|- F Fn _V   =>    |- ( ( A e. V /\ B e. W ) -> ( rec ( F , A ) ` B ) e. _V )
Theorem	rdgifnon 6182	The recursive definition generator is a function on ordinal numbers. The F Fn _V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6189; in cases like df-oadd 6223 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
|- ( ( F Fn _V /\ A e. V ) -> rec ( F , A ) Fn On )
Theorem	rdgifnon2 6183*	The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.)
|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> rec ( F , A ) Fn On )
Theorem	rdgivallem 6184*	Value of the recursive definition generator. Lemma for rdgival 6185 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)
|- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( ( rec ( F , A ) |` B ) ` x ) ) ) )
Theorem	rdgival 6185*	Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.)
|- ( ( F Fn _V /\ A e. V /\ B e. On ) -> ( rec ( F , A ) ` B ) = ( A u. U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) ) )
Theorem	rdgss 6186	Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> I e. V )   &    |- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> ( rec ( F , I ) ` A ) C_ ( rec ( F , I ) ` B ) )
Theorem	rdgisuc1 6187*	One way of describing the value of the recursive definition generator at a successor. There is no condition on the characteristic function F other than F Fn _V. Given that, the resulting expression encompasses both the expected successor term ( F ` ( rec ( F , A ) ` B ) ) but also terms that correspond to the initial value A and to limit ordinals U_ x e. B ( F ` ( rec ( F , A ) ` x ) ). If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 6188. (Contributed by Jim Kingdon, 9-Jun-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. On )   =>    |- ( ph -> ( rec ( F , A ) ` suc B ) = ( A u. ( U_ x e. B ( F ` ( rec ( F , A ) ` x ) ) u. ( F ` ( rec ( F , A ) ` B ) ) ) ) )
Theorem	rdgisucinc 6188*	Value of the recursive definition generator at a successor. This can be thought of as a generalization of oasuc 6265 and omsuc 6273. (Contributed by Jim Kingdon, 29-Aug-2019.)
|- ( ph -> F Fn _V )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. On )   &    |- ( ph -> A. x x C_ ( F ` x ) )   =>    |- ( ph -> ( rec ( F , A ) ` suc B ) = ( F ` ( rec ( F , A ) ` B ) ) )
Theorem	rdgon 6189*	Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)
|- ( ph -> A e. On )   &    |- ( ph -> A. x e. On ( F ` x ) e. On )   =>    |- ( ( ph /\ B e. On ) -> ( rec ( F , A ) ` B ) e. On )
Theorem	rdg0 6190	The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
|- A e. _V   =>    |- ( rec ( F , A ) ` (/) ) = A
Theorem	rdg0g 6191	The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)
|- ( A e. C -> ( rec ( F , A ) ` (/) ) = A )
Theorem	rdgexg 6192	The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.)
|- A e. _V   &    |- F Fn _V   =>    |- ( B e. V -> ( rec ( F , A ) ` B ) e. _V )
2.6.21  Finite recursion
Syntax	cfrec 6193	Extend class notation with the finite recursive definition generator, with characteristic function F and initial value I.
class frec ( F , I )
Definition	df-frec 6194*	Define a recursive definition generator on om (the class of finite ordinals) with characteristic function F and initial value I. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our frec operation (especially when df-recs 6108 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple; see frec0g 6200 and frecsuc 6210. Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4447. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 6211, this definition and df-irdg 6173 restricted to om produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.)
|- frec ( F , I ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. I ) ) } ) ) |` om )
Theorem	freceq1 6195	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- ( F = G -> frec ( F , A ) = frec ( G , A ) )
Theorem	freceq2 6196	Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- ( A = B -> frec ( F , A ) = frec ( F , B ) )
Theorem	frecex 6197	Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.)
|- frec ( F , A ) e. _V
Theorem	frecfun 6198	Finite recursion produces a function. See also frecfnom 6204 which also states that the domain of that function is om but which puts conditions on A and F. (Contributed by Jim Kingdon, 13-Feb-2022.)
|- Fun frec ( F , A )
Theorem	nffrec 6199	Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.)
|- F/_ x F   &    |- F/_ x A   =>    |- F/_ xfrec ( F , A )
Theorem	frec0g 6200	The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)
|- ( A e. V -> (frec ( F , A ) ` (/) ) = A )