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Theorem rdgfun 6022
Description: The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
rdgfun  |-  Fun  rec ( F ,  A )

Proof of Theorem rdgfun
Dummy variables  x  y  z  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2082 . . 3  |-  { f  |  E. y  e.  On  ( f  Fn  y  /\  A. z  e.  y  ( f `  z )  =  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  ( f  |`  z ) ) ) }  =  { f  |  E. y  e.  On  ( f  Fn  y  /\  A. z  e.  y  ( f `  z )  =  ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) `  ( f  |`  z ) ) ) }
21tfrlem7 5966 . 2  |-  Fun recs (
( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )
3 df-irdg 6019 . . 3  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
43funeqi 4952 . 2  |-  ( Fun 
rec ( F ,  A )  <->  Fun recs ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) ) )
52, 4mpbir 144 1  |-  Fun  rec ( F ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285   {cab 2068   A.wral 2349   E.wrex 2350   _Vcvv 2602    u. cun 2972   U_ciun 3686    |-> cmpt 3847   Oncon0 4126   dom cdm 4371    |` cres 4373   Fun wfun 4926    Fn wfn 4927   ` cfv 4932  recscrecs 5953   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-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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  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-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  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-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-recs 5954  df-irdg 6019
This theorem is referenced by:  rdgivallem  6030
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