MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-recs Unicode version

Definition df-recs 6404
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 6439 for more details on why this definition is desirable. Unlike df-rdg 6439 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 6432 and recsval 6433 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7263, zorn2 8149, and dfac8alem 7672. (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 6403 . 2  class recs ( F )
3 vf . . . . . . . 8  set  f
43cv 1631 . . . . . . 7  class  f
5 vx . . . . . . . 8  set  x
65cv 1631 . . . . . . 7  class  x
74, 6wfn 5266 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  set  y
98cv 1631 . . . . . . . . 9  class  y
109, 4cfv 5271 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4707 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 5271 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1632 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2556 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 358 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4408 . . . . 5  class  On
1715, 5, 16wrex 2557 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2282 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3843 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1632 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  6405  nfrecs  6406  recsfval  6413  tfrlem9  6417  dfrdg2  24223
  Copyright terms: Public domain W3C validator