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Theorem tfr1on 6050
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 2085 . . 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 6038 . 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 6049 1  |-  ( ph  ->  Y  C_  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 922    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   _Vcvv 2614    C_ wss 2986   U.cuni 3630   Ord word 4156   suc csuc 4159   dom cdm 4404    |` cres 4406   Fun wfun 4966    Fn wfn 4967   ` cfv 4972  recscrecs 6004
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-suc 4165  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-recs 6005
This theorem is referenced by:  tfri1dALT  6051
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