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Definition df-recs 6305
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 6370 for more details on why this definition is desirable. Unlike df-irdg 6370 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 6335 and tfri2d 6336 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 6304 . 2  class recs ( F )
3 vf . . . . . . . 8  setvar  f
43cv 1352 . . . . . . 7  class  f
5 vx . . . . . . . 8  setvar  x
65cv 1352 . . . . . . 7  class  x
74, 6wfn 5211 . . . . . 6  wff  f  Fn  x
8 vy . . . . . . . . . 10  setvar  y
98cv 1352 . . . . . . . . 9  class  y
109, 4cfv 5216 . . . . . . . 8  class  ( f `
 y )
114, 9cres 4628 . . . . . . . . 9  class  ( f  |`  y )
1211, 1cfv 5216 . . . . . . . 8  class  ( F `
 ( f  |`  y ) )
1310, 12wceq 1353 . . . . . . 7  wff  ( f `
 y )  =  ( F `  (
f  |`  y ) )
1413, 8, 6wral 2455 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )
157, 14wa 104 . . . . 5  wff  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) )
16 con0 4363 . . . . 5  class  On
1715, 5, 16wrex 2456 . . . 4  wff  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )
1817, 3cab 2163 . . 3  class  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
1918cuni 3809 . 2  class  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
202, 19wceq 1353 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  6306  nfrecs  6307  recsfval  6315  tfrlem9  6319  tfr0dm  6322  tfr1onlemssrecs  6339  tfrcllemssrecs  6352
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