Definition df-recs 6391

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 6456 for more details on why this definition is desirable. Unlike df-irdg 6456 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 6421 and tfri2d 6422 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 6390	. 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 5266	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1372	. . . . . . . . 9 class y
10	9, 4	cfv 5271	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4677	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5271	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1373	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2484	. . . . . 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 4410	. . . . 5 class On
17	15, 5, 16	wrex 2485	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2191	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3850	. 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  6392  nfrecs  6393  recsfval  6401  tfrlem9  6405  tfr0dm  6408  tfr1onlemssrecs  6425  tfrcllemssrecs  6438