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Definition df-recs 6024
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 6089 for more details on why this definition is desirable. Unlike df-irdg 6089 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 6054 and tfri2d 6055 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
StepHypRef Expression
1 cF . . 3  class  F
21crecs 6023 . 2  class recs ( F )
3 vf . . . . . . . 8  setvar  f
43cv 1286 . . . . . . 7  class  f
5 vx . . . . . . . 8  setvar  x
65cv 1286 . . . . . . 7  class  x
74, 6wfn 4976 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  setvar  y
98cv 1286 . . . . . . . . 9  class  y
109, 4cfv 4981 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4413 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 4981 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1287 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2355 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 102 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4164 . . . . 5  class  On
1715, 5, 16wrex 2356 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2071 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3636 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1287 1  wff recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  recseq  6025  nfrecs  6026  recsfval  6034  tfrlem9  6038  tfr0dm  6041  tfr1onlemssrecs  6058  tfrcllemssrecs  6071
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