Definition df-recs 6409

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 6474 for more details on why this definition is desirable. Unlike df-irdg 6474 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 6439 and tfri2d 6440 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 6408	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1372	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1372	. . . . . . 7 class x
7	4, 6	wfn 5280	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1372	. . . . . . . . 9 class y
10	9, 4	cfv 5285	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4690	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5285	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1373	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2485	. . . . . 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 4423	. . . . 5 class On
17	15, 5, 16	wrex 2486	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2192	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3859	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1373	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  6410  nfrecs  6411  recsfval  6419  tfrlem9  6423  tfr0dm  6426  tfr1onlemssrecs  6443  tfrcllemssrecs  6456