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Theorem recsfval 6644
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 6635 . 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 4027 . 2  |-  U. A  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }
41, 3eqtr4i 2461 1  |- recs ( F )  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653   {cab 2424   A.wral 2707   E.wrex 2708   U.cuni 4017   Oncon0 4583    |` cres 4882    Fn wfn 5451   ` cfv 5456  recscrecs 6634
This theorem is referenced by:  tfrlem6  6645  tfrlem7  6646  tfrlem8  6647  tfrlem9  6648  tfrlem9a  6649  tfrlem13  6653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-uni 4018  df-recs 6635
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