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Theorem rdgifnon 6355
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 6362; in cases like df-oadd 6396 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 6346 . 2  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
2 rdgruledefgg 6351 . . 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 1867 . 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 6311 1  |-  ( ( F  Fn  _V  /\  A  e.  V )  ->  rec ( F ,  A )  Fn  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2141   _Vcvv 2730    u. cun 3119   U_ciun 3871    |-> cmpt 4048   Oncon0 4346   dom cdm 4609   Fun wfun 5190    Fn wfn 5191   ` cfv 5196   reccrdg 6345
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-recs 6281  df-irdg 6346
This theorem is referenced by:  rdgivallem  6357
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