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Theorem tfri2d 6083
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 6112). 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 6082 . . . . 5  |-  ( ph  ->  F  Fn  On )
4 fndm 5099 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4syl 14 . . . 4  |-  ( ph  ->  dom  F  =  On )
65eleq2d 2157 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  On ) )
76biimpar 291 . 2  |-  ( (
ph  /\  A  e.  On )  ->  A  e. 
dom  F )
81tfr2a 6068 . 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 102   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   Oncon0 4181   dom cdm 4428    |` cres 4430   Fun wfun 4996    Fn wfn 4997   ` cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-recs 6052
This theorem is referenced by:  rdgivallem  6128
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