Theorem tfrlem6 6587

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 4900	. . 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 6584	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5394	. . . 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 6586	. . 3 |- recs ( F ) = U. A
8	7	releqi 4858	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 146	1 |- Rel recs ( F )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   U.cuni 3935   Oncon0 4508   |` cres 4776   Rel wrel 4779   Fun wfun 5371   Fn wfn 5372   ` cfv 5377  recscrecs 6575

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-recs 6576

This theorem is used by:  tfrlem7  6588