Theorem tfrlem8 6579

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 6572	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
3	2	abeq2i 2349	. . . . . . 7 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4		fndm 5475	. . . . . . . . . . 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 2307	. . . . . . . . 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 2662	. . . . . . 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 2310	. . . . . 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 2662	. . . 4 |- ( E. g e. A z = dom g -> z e. On )
13	12	abssi 3323	. . 3 |- { z | E. g e. A z = dom g } C_ On
14		ssorduni 4629	. . 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 6576	. . . . 5 |- recs ( F ) = U. A
17	16	dmeqi 4977	. . . 4 |- dom recs ( F ) = dom U. A
18		dmuni 4986	. . . 4 |- dom U. A = U_ g e. A dom g
19		vex 2824	. . . . . 6 |- g e. _V
20	19	dmex 5044	. . . . 5 |- dom g e. _V
21	20	dfiun2 4041	. . . 4 |- U_ g e. A dom g = U. { z | E. g e. A z = dom g }
22	17, 18, 21	3eqtri 2263	. . 3 |- dom recs ( F ) = U. { z | E. g e. A z = dom g }
23		ordeq 4512	. . 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 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   C_ wss 3220   U.cuni 3930   U_ciun 4007   Ord word 4502   Oncon0 4503   dom cdm 4769   |` cres 4771   Fn wfn 5367   ` cfv 5372  recscrecs 6565

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-tr 4225  df-iord 4506  df-on 4508  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-recs 6566

This theorem is referenced by:  tfrlemi14d  6594  tfri1dALT  6612