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Theorem tfr1on 6213
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 2115 . . 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 6201 . 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 6212 1  |-  ( ph  ->  Y  C_  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   _Vcvv 2658    C_ wss 3039   U.cuni 3704   Ord word 4252   suc csuc 4255   dom cdm 4507    |` cres 4509   Fun wfun 5085    Fn wfn 5086   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfri1dALT  6214
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