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Theorem rdgfun 6352
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 2170 . . 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 6296 . 2  |-  Fun recs (
( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) )
3 df-irdg 6349 . . 3  |-  rec ( F ,  A )  = recs ( ( g  e. 
_V  |->  ( A  u.  U_ x  e.  dom  g
( F `  (
g `  x )
) ) ) )
43funeqi 5219 . 2  |-  ( Fun 
rec ( F ,  A )  <->  Fun recs ( ( g  e.  _V  |->  ( A  u.  U_ x  e.  dom  g ( F `
 ( g `  x ) ) ) ) ) )
52, 4mpbir 145 1  |-  Fun  rec ( F ,  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   {cab 2156   A.wral 2448   E.wrex 2449   _Vcvv 2730    u. cun 3119   U_ciun 3873    |-> cmpt 4050   Oncon0 4348   dom cdm 4611    |` cres 4613   Fun wfun 5192    Fn wfn 5193   ` cfv 5198  recscrecs 6283   reccrdg 6348
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-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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  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-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-recs 6284  df-irdg 6349
This theorem is referenced by:  rdgivallem  6360
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