Theorem tfrlem8 6297

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 6290	. . . . . . . 8 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }
3	2	abeq2i 2281	. . . . . . 7 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4		fndm 5297	. . . . . . . . . . 11 |- ( g Fn z -> dom g = z )
5	4	adantr 274	. . . . . . . . . 10 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g = z )
6	5	eleq1d 2239	. . . . . . . . 9 |- ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> ( dom g e. On <-> z e. On ) )
7	6	biimprcd 159	. . . . . . . 8 |- ( z e. On -> ( ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) -> dom g e. On ) )
8	7	rexlimiv 2581	. . . . . . 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 120	. . . . . 6 |- ( g e. A -> dom g e. On )
10		eleq1a 2242	. . . . . 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 2581	. . . 4 |- ( E. g e. A z = dom g -> z e. On )
13	12	abssi 3222	. . 3 |- { z | E. g e. A z = dom g } C_ On
14		ssorduni 4471	. . 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 6294	. . . . 5 |- recs ( F ) = U. A
17	16	dmeqi 4812	. . . 4 |- dom recs ( F ) = dom U. A
18		dmuni 4821	. . . 4 |- dom U. A = U_ g e. A dom g
19		vex 2733	. . . . . 6 |- g e. _V
20	19	dmex 4877	. . . . 5 |- dom g e. _V
21	20	dfiun2 3907	. . . 4 |- U_ g e. A dom g = U. { z | E. g e. A z = dom g }
22	17, 18, 21	3eqtri 2195	. . 3 |- dom recs ( F ) = U. { z | E. g e. A z = dom g }
23		ordeq 4357	. . 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 145	1 |- Ord dom recs ( F )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   C_ wss 3121   U.cuni 3796   U_ciun 3873   Ord word 4347   Oncon0 4348   dom cdm 4611   |` cres 4613   Fn wfn 5193   ` cfv 5198  recscrecs 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-tr 4088  df-iord 4351  df-on 4353  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-recs 6284

This theorem is referenced by:  tfrlemi14d  6312  tfri1dALT  6330