ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-recs Unicode version

Definition df-recs 6002
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-irdg 6067 for more details on why this definition is desirable. Unlike df-irdg 6067 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 tfri1d 6032 and tfri2d 6033 for the primary contract of this definition.

(Contributed by Stefan O'Rear, 18-Jan-2015.)

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 6001 . 2  class recs ( F )
3 vf . . . . . . . 8  setvar  f
43cv 1284 . . . . . . 7  class  f
5 vx . . . . . . . 8  setvar  x
65cv 1284 . . . . . . 7  class  x
74, 6wfn 4964 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  setvar  y
98cv 1284 . . . . . . . . 9  class  y
109, 4cfv 4969 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4403 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 4969 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1285 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2353 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 102 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4154 . . . . 5  class  On
1715, 5, 16wrex 2354 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2069 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3627 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1285 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  6003  nfrecs  6004  recsfval  6012  tfrlem9  6016  tfr0dm  6019  tfr1onlemssrecs  6036  tfrcllemssrecs  6049
  Copyright terms: Public domain W3C validator