Definition df-recs 6202

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 6267 for more details on why this definition is desirable. Unlike df-irdg 6267 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 6232 and tfri2d 6233 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 6201	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1330	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1330	. . . . . . 7 class x
7	4, 6	wfn 5118	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1330	. . . . . . . . 9 class y
10	9, 4	cfv 5123	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4541	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5123	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1331	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2416	. . . . . 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 4285	. . . . 5 class On
17	15, 5, 16	wrex 2417	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2125	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3736	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1331	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  6203  nfrecs  6204  recsfval  6212  tfrlem9  6216  tfr0dm  6219  tfr1onlemssrecs  6236  tfrcllemssrecs  6249