Definition df-recs 6299

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 6364 for more details on why this definition is desirable. Unlike df-irdg 6364 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 6329 and tfri2d 6330 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 6298	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1352	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1352	. . . . . . 7 class x
7	4, 6	wfn 5206	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1352	. . . . . . . . 9 class y
10	9, 4	cfv 5211	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4624	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5211	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1353	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2455	. . . . . 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 4359	. . . . 5 class On
17	15, 5, 16	wrex 2456	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2163	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3807	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1353	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  6300  nfrecs  6301  recsfval  6309  tfrlem9  6313  tfr0dm  6316  tfr1onlemssrecs  6333  tfrcllemssrecs  6346