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Theorem tfri2d 5981
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 5982). Here we show that the function  F has the property that for any function  G satisfying that condition, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by Jim Kingdon, 4-May-2019.)
Hypotheses
Ref Expression
tfri1d.1  |-  F  = recs ( G )
tfri1d.2  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
Assertion
Ref Expression
tfri2d  |-  ( (
ph  /\  A  e.  On )  ->  ( F `
 A )  =  ( G `  ( F  |`  A ) ) )
Distinct variable group:    x, G
Allowed substitution hints:    ph( x)    A( x)    F( x)

Proof of Theorem tfri2d
StepHypRef Expression
1 tfri1d.1 . . . . . 6  |-  F  = recs ( G )
2 tfri1d.2 . . . . . 6  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
31, 2tfri1d 5980 . . . . 5  |-  ( ph  ->  F  Fn  On )
4 fndm 5026 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4syl 14 . . . 4  |-  ( ph  ->  dom  F  =  On )
65eleq2d 2123 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  On ) )
76biimpar 285 . 2  |-  ( (
ph  /\  A  e.  On )  ->  A  e. 
dom  F )
81tfr2a 5967 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
97, 8syl 14 1  |-  ( (
ph  /\  A  e.  On )  ->  ( F `
 A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574   Oncon0 4128   dom cdm 4373    |` cres 4375   Fun wfun 4924    Fn wfn 4925   ` cfv 4930  recscrecs 5950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951
This theorem is referenced by:  rdgivallem  5999  frecsuclem1  6018
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