Theorem tfrlem8 6549

Description: Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)

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
tfrlem8	|- Ord dom recs ( F )

Distinct variable group:   x, f, y, F

Allowed substitution hints:   A( x, y, f)

Proof of Theorem tfrlem8

Dummy variables g z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . . . . . . 9 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem3 6542	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
3	2	abeq2i 2343	. . . . . . 7 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4		fndm 5455	. . . . . . . . . . 11 |- ( g Fn z -> dom g = z )
5	4	adantr 276	. . . . . . . . . 10 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g = z )
6	5	eleq1d 2301	. . . . . . . . 9 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> ( dom g e. On <-> z e. On ) )
7	6	biimprcd 160	. . . . . . . 8 |- ( z e. On -> ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g e. On ) )
8	7	rexlimiv 2654	. . . . . . 7 |- ( E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g e. On )
9	3, 8	sylbi 121	. . . . . 6 |- ( g e. A -> dom g e. On )
10		eleq1a 2304	. . . . . 6 |- ( dom g e. On -> ( z = dom g -> z e. On ) )
11	9, 10	syl 14	. . . . 5 |- ( g e. A -> ( z = dom g -> z e. On ) )
12	11	rexlimiv 2654	. . . 4 |- ( E. g e. A z = dom g -> z e. On )
13	12	abssi 3313	. . 3 |- { z | E. g e. A z = dom g } C_ On
14		ssorduni 4609	. . 3 |- ( { z | E. g e. A z = dom g } C_ On -> Ord U. { z | E. g e. A z = dom g } )
15	13, 14	ax-mp 5	. 2 |- Ord U. { z | E. g e. A z = dom g }
16	1	recsfval 6546	. . . . 5 |- recs ( F ) = U. A
17	16	dmeqi 4957	. . . 4 |- dom recs ( F ) = dom U. A
18		dmuni 4966	. . . 4 |- dom U. A = U_ g e. A dom g
19		vex 2816	. . . . . 6 |- g e. _V
20	19	dmex 5024	. . . . 5 |- dom g e. _V
21	20	dfiun2 4025	. . . 4 |- U_ g e. A dom g = U. { z | E. g e. A z = dom g }
22	17, 18, 21	3eqtri 2257	. . 3 |- dom recs ( F ) = U. { z | E. g e. A z = dom g }
23		ordeq 4493	. . 3 |- ( dom recs ( F ) = U. { z | E. g e. A z = dom g } -> ( Ord dom recs ( F ) <-> Ord U. { z | E. g e. A z = dom g } ) )
24	22, 23	ax-mp 5	. 2 |- ( Ord dom recs ( F ) <-> Ord U. { z | E. g e. A z = dom g } )
25	15, 24	mpbir 146	1 |- Ord dom recs ( F )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   C_ wss 3211   U.cuni 3914   U_ciun 3991   Ord word 4483   Oncon0 4484   dom cdm 4749   |` cres 4751   Fn wfn 5347   ` cfv 5352  recscrecs 6535

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-tr 4209  df-iord 4487  df-on 4489  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-recs 6536

This theorem is referenced by:  tfrlemi14d  6564  tfri1dALT  6582