Definition df-recs 6273

Description: 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 6338 for more details on why this definition is desirable. Unlike df-irdg 6338 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 6303 and tfri2d 6304 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

Assertion

Ref	Expression
df-recs	|- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Distinct variable group:   f, F, x, y

Detailed syntax breakdown of Definition df-recs

Step	Hyp	Ref	Expression
1		cF	. . 3 class F
2	1	crecs 6272	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1342	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1342	. . . . . . 7 class x
7	4, 6	wfn 5183	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1342	. . . . . . . . 9 class y
10	9, 4	cfv 5188	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4606	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5188	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1343	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2444	. . . . . 6 wff A. y e. x ( f ` y ) = ( F ` ( f |` y ) )
15	7, 14	wa 103	. . . . 5 wff ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
16		con0 4341	. . . . 5 class On
17	15, 5, 16	wrex 2445	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2151	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3789	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1343	1 wff recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

This definition is referenced by:  recseq  6274  nfrecs  6275  recsfval  6283  tfrlem9  6287  tfr0dm  6290  tfr1onlemssrecs  6307  tfrcllemssrecs  6320