Theorem tfrlem6 6580

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 4898	. . 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 6577	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5392	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 14	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2608	. 2 |- Rel U. A
7	2	recsfval 6579	. . 3 |- recs ( F ) = U. A
8	7	releqi 4856	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 146	1 |- Rel recs ( F )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   U.cuni 3933   Oncon0 4506   |` cres 4774   Rel wrel 4777   Fun wfun 5369   Fn wfn 5370   ` cfv 5375  recscrecs 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-recs 6569

This theorem is referenced by:  tfrlem7  6581