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Theorem tfrlem6 6081
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 4560 . . 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 6078 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5032 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 14 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2433 . 2  |-  Rel  U. A
72recsfval 6080 . . 3  |- recs ( F )  =  U. A
87releqi 4521 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 144 1  |-  Rel recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   U.cuni 3653   Oncon0 4190    |` cres 4440   Rel wrel 4443   Fun wfun 5009    Fn wfn 5010   ` cfv 5015  recscrecs 6069
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-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-recs 6070
This theorem is referenced by:  tfrlem7  6082
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