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Theorem tfrlem6 7043
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 5118 . . 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 7040 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5598 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 16 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2823 . 2  |-  Rel  U. A
72recsfval 7042 . . 3  |- recs ( F )  =  U. A
87releqi 5079 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 209 1  |-  Rel recs ( F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   U.cuni 4240   Oncon0 4873    |` cres 4996   Rel wrel 4999   Fun wfun 5575    Fn wfn 5576   ` cfv 5581  recscrecs 7033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589  df-recs 7034
This theorem is referenced by:  tfrlem7  7044  tfrlem11  7049  tfrlem15  7053  tfrlem16  7054
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