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Theorem tfrlem6 6606
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 4960 . . 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 6603 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5434 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 16 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2740 . 2  |-  Rel  U. A
72recsfval 6605 . . 3  |- recs ( F )  =  U. A
87releqi 4923 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 201 1  |-  Rel recs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2394   A.wral 2670   E.wrex 2671   U.cuni 3979   Oncon0 4545    |` cres 4843   Rel wrel 4846   Fun wfun 5411    Fn wfn 5412   ` cfv 5417  recscrecs 6595
This theorem is referenced by:  tfrlem7  6607  tfrlem11  6612  tfrlem15  6616  tfrlem16  6617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-res 4853  df-iota 5381  df-fun 5419  df-fn 5420  df-fv 5425  df-recs 6596
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