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Theorem rdgifnon 6028
Description: The recursive definition generator is a function on ordinal numbers. The  F  Fn  _V condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 6035; in cases like df-oadd 6069 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.)
Assertion
Ref Expression
rdgifnon  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )

Proof of Theorem rdgifnon
Dummy variables  f  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-irdg 6019 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2 rdgruledefgg 6024 . . 3  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  ( Fun  ( g  e.  _V  |->  ( A  u.  U_ x  e. 
dom  g ( F `
 ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  f )  e.  _V ) )
32alrimiv 1796 . 2  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  A. f ( Fun  ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) )  /\  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  f )  e.  _V ) )
41, 3tfri1d 5984 1  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   _Vcvv 2602    u. cun 2972   U_ciun 3686    |-> cmpt 3847   Oncon0 4126   dom cdm 4371   Fun wfun 4926    Fn wfn 4927   ` cfv 4932   reccrdg 6018
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-recs 5954  df-irdg 6019
This theorem is referenced by:  rdgivallem  6030
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