Definition df-recs 6390

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 6455 for more details on why this definition is desirable. Unlike df-irdg 6455 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 6420 and tfri2d 6421 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 6389	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1371	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1371	. . . . . . 7 class x
7	4, 6	wfn 5265	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1371	. . . . . . . . 9 class y
10	9, 4	cfv 5270	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4676	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5270	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1372	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2483	. . . . . 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 4409	. . . . 5 class On
17	15, 5, 16	wrex 2484	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2190	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3849	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1372	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  6391  nfrecs  6392  recsfval  6400  tfrlem9  6404  tfr0dm  6407  tfr1onlemssrecs  6424  tfrcllemssrecs  6437