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Theorem recsfval 6180
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 6170 . 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 3716 . 2  |-  U. A  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }
41, 3eqtr4i 2141 1  |- recs ( F )  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316   {cab 2103   A.wral 2393   E.wrex 2394   U.cuni 3706   Oncon0 4255    |` cres 4511    Fn wfn 5088   ` cfv 5093  recscrecs 6169
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-uni 3707  df-recs 6170
This theorem is referenced by:  tfrlem6  6181  tfrlem7  6182  tfrlem8  6183  tfrlem9  6184  tfrlemibfn  6193  tfrlemiubacc  6195  tfrlemi14d  6198  tfrexlem  6199
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