Definition df-recs 6566

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 6631 for more details on why this definition is desirable. Unlike df-irdg 6631 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 6596 and tfri2d 6597 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 6565	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1401	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1401	. . . . . . 7 class x
7	4, 6	wfn 5367	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1401	. . . . . . . . 9 class y
10	9, 4	cfv 5372	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4771	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5372	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1402	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2528	. . . . . 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 4503	. . . . 5 class On
17	15, 5, 16	wrex 2529	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2224	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3930	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1402	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  6567  nfrecs  6568  recsfval  6576  tfrlem9  6580  tfr0dm  6583  tfr1onlemssrecs  6600  tfrcllemssrecs  6613