Definition df-recs 6449

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 6514 for more details on why this definition is desirable. Unlike df-irdg 6514 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 6479 and tfri2d 6480 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 6448	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1394	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1394	. . . . . . 7 class x
7	4, 6	wfn 5312	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1394	. . . . . . . . 9 class y
10	9, 4	cfv 5317	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4720	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5317	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1395	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2508	. . . . . 6 wff A. y e. x ( f ` y ) = ( F ` ( f |` y ) )
15	7, 14	wa 104	. . . . 5 wff ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
16		con0 4453	. . . . 5 class On
17	15, 5, 16	wrex 2509	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2215	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3887	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1395	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  6450  nfrecs  6451  recsfval  6459  tfrlem9  6463  tfr0dm  6466  tfr1onlemssrecs  6483  tfrcllemssrecs  6496