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Theorem tfr1on 6097
Description: Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f  |-  F  = recs ( G )
tfr1on.g  |-  ( ph  ->  Fun  G )
tfr1on.x  |-  ( ph  ->  Ord  X )
tfr1on.ex  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
tfr1on.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfr1on.yx  |-  ( ph  ->  Y  e.  X )
Assertion
Ref Expression
tfr1on  |-  ( ph  ->  Y  C_  dom  F )
Distinct variable groups:    f, G, x   
f, X, x    f, Y, x    ph, f, x
Allowed substitution hints:    F( x, f)

Proof of Theorem tfr1on
Dummy variables  a  b  c  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfr1on.f . 2  |-  F  = recs ( G )
2 tfr1on.g . 2  |-  ( ph  ->  Fun  G )
3 tfr1on.x . 2  |-  ( ph  ->  Ord  X )
4 tfr1on.ex . 2  |-  ( (
ph  /\  x  e.  X  /\  f  Fn  x
)  ->  ( G `  f )  e.  _V )
5 eqid 2088 . . 3  |-  { a  |  E. b  e.  X  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }  =  { a  |  E. b  e.  X  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }
65tfr1onlem3 6085 . 2  |-  { a  |  E. b  e.  X  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( G `  ( a  |`  c ) ) ) }  =  { f  |  E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
7 tfr1on.u . 2  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
8 tfr1on.yx . 2  |-  ( ph  ->  Y  e.  X )
91, 2, 3, 4, 6, 7, 8tfr1onlemres 6096 1  |-  ( ph  ->  Y  C_  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    C_ wss 2997   U.cuni 3648   Ord word 4180   suc csuc 4183   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:  tfri1dALT  6098
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