Definition df-recs 6132

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 6197 for more details on why this definition is desirable. Unlike df-irdg 6197 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 6162 and tfri2d 6163 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 6131	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1298	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1298	. . . . . . 7 class x
7	4, 6	wfn 5054	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1298	. . . . . . . . 9 class y
10	9, 4	cfv 5059	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4479	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5059	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1299	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2375	. . . . . 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 4223	. . . . 5 class On
17	15, 5, 16	wrex 2376	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2086	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3683	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1299	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  6133  nfrecs  6134  recsfval  6142  tfrlem9  6146  tfr0dm  6149  tfr1onlemssrecs  6166  tfrcllemssrecs  6179