Theorem tfrlem6 6221

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.

Step	Hyp	Ref	Expression
1		reluni 4670	. . 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 ) ) ) }
3	2	tfrlem4 6218	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5148	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 14	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2493	. 2 |- Rel U. A
7	2	recsfval 6220	. . 3 |- recs ( F ) = U. A
8	7	releqi 4630	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 145	1 |- Rel recs ( F )

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   U.cuni 3744   Oncon0 4293   |` cres 4549   Rel wrel 4552   Fun wfun 5125   Fn wfn 5126   ` cfv 5131  recscrecs 6209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfrlem7  6222