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Theorem tfrlem6 6646
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 5000 . . 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 6643 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5474 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 16 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2778 . 2  |-  Rel  U. A
72recsfval 6645 . . 3  |- recs ( F )  =  U. A
87releqi 4963 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 202 1  |-  Rel recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   U.cuni 4017   Oncon0 4584    |` cres 4883   Rel wrel 4886   Fun wfun 5451    Fn wfn 5452   ` cfv 5457  recscrecs 6635
This theorem is referenced by:  tfrlem7  6647  tfrlem11  6652  tfrlem15  6656  tfrlem16  6657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-3an 939  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-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-recs 6636
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