Definition df-recs 6536

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 6601 for more details on why this definition is desirable. Unlike df-irdg 6601 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 6566 and tfri2d 6567 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 6535	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1397	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1397	. . . . . . 7 class x
7	4, 6	wfn 5347	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1397	. . . . . . . . 9 class y
10	9, 4	cfv 5352	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4751	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5352	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1398	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2520	. . . . . 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 4484	. . . . 5 class On
17	15, 5, 16	wrex 2521	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2218	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3914	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1398	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  6537  nfrecs  6538  recsfval  6546  tfrlem9  6550  tfr0dm  6553  tfr1onlemssrecs  6570  tfrcllemssrecs  6583