Definition df-recs 6470

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 6535 for more details on why this definition is desirable. Unlike df-irdg 6535 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 6500 and tfri2d 6501 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 6469	. 2 class recs ( F )
3		vf	. . . . . . . 8 setvar f
4	3	cv 1396	. . . . . . 7 class f
5		vx	. . . . . . . 8 setvar x
6	5	cv 1396	. . . . . . 7 class x
7	4, 6	wfn 5321	. . . . . 6 wff f Fn x
8		vy	. . . . . . . . . 10 setvar y
9	8	cv 1396	. . . . . . . . 9 class y
10	9, 4	cfv 5326	. . . . . . . 8 class ( f ` y )
11	4, 9	cres 4727	. . . . . . . . 9 class ( f |` y )
12	11, 1	cfv 5326	. . . . . . . 8 class ( F ` ( f |` y ) )
13	10, 12	wceq 1397	. . . . . . 7 wff ( f ` y ) = ( F ` ( f |` y ) )
14	13, 8, 6	wral 2510	. . . . . 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 4460	. . . . 5 class On
17	15, 5, 16	wrex 2511	. . . 4 wff E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) )
18	17, 3	cab 2217	. . 3 class { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
19	18	cuni 3893	. 2 class U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
20	2, 19	wceq 1397	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  6471  nfrecs  6472  recsfval  6480  tfrlem9  6484  tfr0dm  6487  tfr1onlemssrecs  6504  tfrcllemssrecs  6517