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Definition df-recs 6596
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 6631 for more details on why this definition is desirable. Unlike df-rdg 6631 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 6624 and recsval 6625 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7461, zorn2 8346, and dfac8alem 7870. (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 6595 . 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 5412 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  set  y
98cv 1648 . . . . . . . . 9  class  y
109, 4cfv 5417 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4843 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 5417 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1649 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2670 . . . . . 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 4545 . . . . 5  class  On
1715, 5, 16wrex 2671 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2394 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3979 . 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  6597  nfrecs  6598  recsfval  6605  tfrlem9  6609  dfrdg2  25370
  Copyright terms: Public domain W3C validator