Definition df-recs 5943

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 5980 for more details on why this definition is desirable. Unlike df-irdg 5980 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 5972 and tfri2d 5973 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 5942	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1283	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1283	. . . . . . 7 class x
7	4, 6	wfn 4917	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1283	. . . . . . . . 9 class y
10	9, 4	cfv 4922	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4365	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 4922	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1284	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2348	. . . . . 6 wff A. y e. x ( f ` y ) = ( F ` ( f |` y ) )
15	7, 14	wa 102	. . . . 5 wff ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
16		con0 4118	. . . . 5 class On
17	15, 5, 16	wrex 2349	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2067	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3601	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1284	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  5944  nfrecs  5945  recsfval  5954  tfrlem9  5958  tfr0  5960