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Definition df-recs 6385
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 6420 for more details on why this definition is desirable. Unlike df-rdg 6420 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 6413 and recsval 6414 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7244, zorn2 8130, and dfac8alem 7653. (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 6384 . 2  class recs ( F )
3 vf . . . . . . . 8  set  f
43cv 1624 . . . . . . 7  class  f
5 vx . . . . . . . 8  set  x
65cv 1624 . . . . . . 7  class  x
74, 6wfn 5218 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  set  y
98cv 1624 . . . . . . . . 9  class  y
109, 4cfv 5223 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4692 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 5223 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1625 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2546 . . . . . 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 4393 . . . . 5  class  On
1715, 5, 16wrex 2547 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2272 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3830 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1625 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  6386  nfrecs  6387  recsfval  6394  tfrlem9  6398  dfrdg2  23555
  Copyright terms: Public domain W3C validator