Definition df-recs 6306

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 6371 for more details on why this definition is desirable. Unlike df-irdg 6371 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 6336 and tfri2d 6337 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 6305	. 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 5212	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1352	. . . . . . . . 9 class y
10	9, 4	cfv 5217	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4629	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5217	. . . . . . . 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 4364	. . . . 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 3810	. 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  6307  nfrecs  6308  recsfval  6316  tfrlem9  6320  tfr0dm  6323  tfr1onlemssrecs  6340  tfrcllemssrecs  6353