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Theorem recsfval 6062
Description: Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
recsfval  |- recs ( F )  =  U. A
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem recsfval
StepHypRef Expression
1 df-recs 6052 . 2  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
2 tfrlem.1 . . 3  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
32unieqi 3658 . 2  |-  U. A  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }
41, 3eqtr4i 2111 1  |- recs ( F )  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   {cab 2074   A.wral 2359   E.wrex 2360   U.cuni 3648   Oncon0 4181    |` cres 4430    Fn wfn 4997   ` cfv 5002  recscrecs 6051
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649  df-recs 6052
This theorem is referenced by:  tfrlem6  6063  tfrlem7  6064  tfrlem8  6065  tfrlem9  6066  tfrlemibfn  6075  tfrlemiubacc  6077  tfrlemi14d  6080  tfrexlem  6081
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