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Theorem tfrlem6 7105
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 4959 . . 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 7102 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5602 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 17 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2754 . 2  |-  Rel  U. A
72recsfval 7104 . . 3  |- recs ( F )  =  U. A
87releqi 4921 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 213 1  |-  Rel recs ( F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1446    e. wcel 1889   {cab 2439   A.wral 2739   E.wrex 2740   U.cuni 4201    |` cres 4839   Rel wrel 4842   Oncon0 5426   Fun wfun 5579    Fn wfn 5580   ` cfv 5585  recscrecs 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-wrecs 7033  df-recs 7095
This theorem is referenced by:  tfrlem7  7106  tfrlem11  7111  tfrlem15  7115  tfrlem16  7116
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