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Definition df-recs 6342
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 6377 for more details on why this definition is desirable. Unlike df-rdg 6377 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 6370 and recsval 6371 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7201, zorn2 8087, and dfac8alem 7610. (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 6341 . 2  class recs ( F )
3 vf . . . . . . . 8  set  f
43cv 1618 . . . . . . 7  class  f
5 vx . . . . . . . 8  set  x
65cv 1618 . . . . . . 7  class  x
74, 6wfn 4654 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  set  y
98cv 1618 . . . . . . . . 9  class  y
109, 4cfv 4659 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4649 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 4659 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1619 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2516 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 360 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4350 . . . . 5  class  On
1715, 5, 16wrex 2517 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2242 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3787 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1619 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  6343  nfrecs  6344  recsfval  6351  tfrlem9  6355  dfrdg2  23507
  Copyright terms: Public domain W3C validator