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Theorem tfrlem6 6540
Description: Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised 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
tfrlem6  |-  Rel recs ( F )
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem6
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 reluni 4911 . . 3  |-  ( Rel  U. A  <->  A. g  e.  A  Rel  g )
2 tfrlem.1 . . . . 5  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
32tfrlem4 6537 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5375 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 15 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2698 . 2  |-  Rel  U. A
72recsfval 6539 . . 3  |- recs ( F )  =  U. A
87releqi 4875 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 200 1  |-  Rel recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1647    e. wcel 1715   {cab 2352   A.wral 2628   E.wrex 2629   U.cuni 3929   Oncon0 4495    |` cres 4794   Rel wrel 4797   Fun wfun 5352    Fn wfn 5353   ` cfv 5358  recscrecs 6529
This theorem is referenced by:  tfrlem7  6541  tfrlem11  6546  tfrlem15  6550  tfrlem16  6551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-fv 5366  df-recs 6530
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