Theorem tfrlem8 5957

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 5949	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
3	2	abeq2i 2189	. . . . . . 7 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4		fndm 5018	. . . . . . . . . . 11 |- ( g Fn z -> dom g = z )
5	4	adantr 270	. . . . . . . . . 10 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g = z )
6	5	eleq1d 2147	. . . . . . . . 9 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> ( dom g e. On <-> z e. On ) )
7	6	biimprcd 158	. . . . . . . 8 |- ( z e. On -> ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g e. On ) )
8	7	rexlimiv 2471	. . . . . . 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 119	. . . . . 6 |- ( g e. A -> dom g e. On )
10		eleq1a 2150	. . . . . 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 2471	. . . 4 |- ( E. g e. A z = dom g -> z e. On )
13	12	abssi 3069	. . 3 |- { z | E. g e. A z = dom g } C_ On
14		ssorduni 4231	. . 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 7	. 2 |- Ord U. { z | E. g e. A z = dom g }
16	1	recsfval 5954	. . . . 5 |- recs ( F ) = U. A
17	16	dmeqi 4554	. . . 4 |- dom recs ( F ) = dom U. A
18		dmuni 4563	. . . 4 |- dom U. A = U_ g e. A dom g
19		vex 2604	. . . . . 6 |- g e. _V
20	19	dmex 4616	. . . . 5 |- dom g e. _V
21	20	dfiun2 3712	. . . 4 |- U_ g e. A dom g = U. { z | E. g e. A z = dom g }
22	17, 18, 21	3eqtri 2105	. . 3 |- dom recs ( F ) = U. { z | E. g e. A z = dom g }
23		ordeq 4127	. . 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 7	. 2 |- ( Ord dom recs ( F ) <-> Ord U. { z | E. g e. A z = dom g } )
25	15, 24	mpbir 144	1 |- Ord dom recs ( F )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   C_ wss 2973   U.cuni 3601   U_ciun 3678   Ord word 4117   Oncon0 4118   dom cdm 4363   |` cres 4365   Fn wfn 4917   ` cfv 4922  recscrecs 5942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-tr 3876  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-recs 5943

This theorem is referenced by:  tfrlemi14d  5970