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Definition df-recs 6569
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-rdg 6604 for more details on why this definition is desirable. Unlike df-rdg 6604 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 recsfnon 6597 and recsval 6598 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7434, zorn2 8319, and dfac8alem 7843. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

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 6568 . 2  class recs ( F )
3 vf . . . . . . . 8  set  f
43cv 1648 . . . . . . 7  class  f
5 vx . . . . . . . 8  set  x
65cv 1648 . . . . . . 7  class  x
74, 6wfn 5389 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  set  y
98cv 1648 . . . . . . . . 9  class  y
109, 4cfv 5394 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4820 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 5394 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1649 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2649 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 359 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4522 . . . . 5  class  On
1715, 5, 16wrex 2650 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2373 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3957 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1649 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  6570  nfrecs  6571  recsfval  6578  tfrlem9  6582  dfrdg2  25176
  Copyright terms: Public domain W3C validator