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Theorem recsfval 6413
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 6404 . 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 3853 . 2  |-  U. A  =  U. { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) }
41, 3eqtr4i 2319 1  |- recs ( F )  =  U. A
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   {cab 2282   A.wral 2556   E.wrex 2557   U.cuni 3843   Oncon0 4408    |` cres 4707    Fn wfn 5266   ` cfv 5271  recscrecs 6403
This theorem is referenced by:  tfrlem6  6414  tfrlem7  6415  tfrlem8  6416  tfrlem9  6417  tfrlem9a  6418  tfrlem13  6422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-uni 3844  df-recs 6404
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