Theorem tfrlem8 6344

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 6337	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
3	2	abeq2i 2300	. . . . . . 7 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4		fndm 5334	. . . . . . . . . . 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 2258	. . . . . . . . 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 2601	. . . . . . 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 2261	. . . . . 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 2601	. . . 4 |- ( E. g e. A z = dom g -> z e. On )
13	12	abssi 3245	. . 3 |- { z | E. g e. A z = dom g } C_ On
14		ssorduni 4504	. . 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 6341	. . . . 5 |- recs ( F ) = U. A
17	16	dmeqi 4846	. . . 4 |- dom recs ( F ) = dom U. A
18		dmuni 4855	. . . 4 |- dom U. A = U_ g e. A dom g
19		vex 2755	. . . . . 6 |- g e. _V
20	19	dmex 4911	. . . . 5 |- dom g e. _V
21	20	dfiun2 3935	. . . 4 |- U_ g e. A dom g = U. { z | E. g e. A z = dom g }
22	17, 18, 21	3eqtri 2214	. . 3 |- dom recs ( F ) = U. { z | E. g e. A z = dom g }
23		ordeq 4390	. . 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 1364   e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   C_ wss 3144   U.cuni 3824   U_ciun 3901   Ord word 4380   Oncon0 4381   dom cdm 4644   |` cres 4646   Fn wfn 5230   ` cfv 5235  recscrecs 6330

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-tr 4117  df-iord 4384  df-on 4386  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-recs 6331

This theorem is referenced by:  tfrlemi14d  6359  tfri1dALT  6377