Theorem tfrlem6 7043

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 5118	. . 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 7040	. . . 4 |- ( g e. A -> Fun g )
4		funrel 5598	. . . 4 |- ( Fun g -> Rel g )
5	3, 4	syl 16	. . 3 |- ( g e. A -> Rel g )
6	1, 5	mprgbir 2823	. 2 |- Rel U. A
7	2	recsfval 7042	. . 3 |- recs ( F ) = U. A
8	7	releqi 5079	. 2 |- ( Rel recs ( F ) <-> Rel U. A )
9	6, 8	mpbir 209	1 |- Rel recs ( F )

Syntax hints:   /\ wa 369   = wceq 1374   e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   U.cuni 4240   Oncon0 4873   |` cres 4996   Rel wrel 4999   Fun wfun 5575   Fn wfn 5576   ` cfv 5581  recscrecs 7033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589  df-recs 7034

This theorem is referenced by:  tfrlem7  7044  tfrlem11  7049  tfrlem15  7053  tfrlem16  7054