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.

Step	Hyp	Ref	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 ) ) ) }
3	2	tfrlem4 7003	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5540	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 17	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2765	. 2 |- Rel U. A
7	2	recsfval 7005	. . 3 |- recs ( F ) = U. A
8	7	releqi 5026	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 209	1 |- Rel recs ( F )

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