Theorem tfrlem6 6213

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 4662	. . 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 6210	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5140	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 14	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2490	. 2 |- Rel U. A
7	2	recsfval 6212	. . 3 |- recs ( F ) = U. A
8	7	releqi 4622	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 145	1 |- Rel recs ( F )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   U.cuni 3736   Oncon0 4285   |` cres 4541   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118   ` cfv 5123  recscrecs 6201

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-recs 6202

This theorem is referenced by:  tfrlem7  6214