ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfr1on Unicode version

Theorem tfr1on 6405
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 2193 . . 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 6393 . 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 6404 1  |-  ( ph  ->  Y  C_  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760    C_ wss 3154   U.cuni 3836   Ord word 4394   suc csuc 4397   dom cdm 4660    |` cres 4662   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  recscrecs 6359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6360
This theorem is referenced by:  tfri1dALT  6406
  Copyright terms: Public domain W3C validator