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Theorem tfrlem6 7006
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 5064 . . 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 7003 . . . 4  |-  ( g  e.  A  ->  Fun  g )
4 funrel 5540 . . . 4  |-  ( Fun  g  ->  Rel  g )
53, 4syl 17 . . 3  |-  ( g  e.  A  ->  Rel  g )
61, 5mprgbir 2765 . 2  |-  Rel  U. A
72recsfval 7005 . . 3  |- recs ( F )  =  U. A
87releqi 5026 . 2  |-  ( Rel recs
( F )  <->  Rel  U. A
)
96, 8mpbir 209 1  |-  Rel recs ( F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1403    e. wcel 1840   {cab 2385   A.wral 2751   E.wrex 2752   U.cuni 4188   Oncon0 4819    |` cres 4942   Rel wrel 4945   Fun wfun 5517    Fn wfn 5518   ` cfv 5523  recscrecs 6996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531  df-recs 6997
This theorem is referenced by:  tfrlem7  7007  tfrlem11  7012  tfrlem15  7016  tfrlem16  7017
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