Theorem tfrlem8 3903

Description: Lemma for transfinite recursion. The domain of F is ordinal. (The proof was shortened by Alan Sare, 11-Mar-2008.)

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A

Assertion

Ref	Expression
tfrlem8	|- Ord dom F

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfrlem8

Step	Hyp	Ref	Expression
1		eleq1a 1535	. . . . . . 7 |- (dom f e. On -> (x = dom f -> x e. On))
2	1	imp 350	. . . . . 6 |- ((dom f e. On /\ x = dom f) -> x e. On)
3		tfrlem.1	. . . . . . . . . 10 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
4	3	abeq2i 1562	. . . . . . . . 9 |- (f e. A <-> E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))))
5		pm3.26 319	. . . . . . . . . 10 |- ((f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) -> f Fn x)
6	5	r19.22si 1726	. . . . . . . . 9 |- (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) -> E.x e. On f Fn x)
7	4, 6	sylbi 199	. . . . . . . 8 |- (f e. A -> E.x e. On f Fn x)
8		df-rex 1642	. . . . . . . 8 |- (E.x e. On f Fn x <-> E.x(x e. On /\ f Fn x))
9	7, 8	sylib 198	. . . . . . 7 |- (f e. A -> E.x(x e. On /\ f Fn x))
10		eleq1a 1535	. . . . . . . . . 10 |- (x e. On -> (dom f = x -> dom f e. On))
11	10	imp 350	. . . . . . . . 9 |- ((x e. On /\ dom f = x) -> dom f e. On)
12		fndm 3573	. . . . . . . . 9 |- (f Fn x -> dom f = x)
13	11, 12	sylan2 451	. . . . . . . 8 |- ((x e. On /\ f Fn x) -> dom f e. On)
14	13	19.23aiv 1290	. . . . . . 7 |- (E.x(x e. On /\ f Fn x) -> dom f e. On)
15	9, 14	syl 10	. . . . . 6 |- (f e. A -> dom f e. On)
16	2, 15	sylan 448	. . . . 5 |- ((f e. A /\ x = dom f) -> x e. On)
17	16	r19.23aiva 1736	. . . 4 |- (E.f e. A x = dom f -> x e. On)
18	17	abssi 2112	. . 3 |- {x | E.f e. A x = dom f} (_ On
19		ssorduni 2983	. . 3 |- ({x | E.f e. A x = dom f} (_ On -> Ord U.{x | E.f e. A x = dom f})
20	18, 19	ax-mp 7	. 2 |- Ord U.{x | E.f e. A x = dom f}
21		tfrlem.2	. . . . 5 |- F = U.A
22	21	dmeqi 3301	. . . 4 |- dom F = dom U. A
23		dmuni 3308	. . . 4 |- dom U. A = U_f e. A dom f
24		visset 1804	. . . . . 6 |- f e. V
25		dmexg 3344	. . . . . 6 |- (f e. V -> dom f e. V)
26	24, 25	ax-mp 7	. . . . 5 |- dom f e. V
27	26	dfiun2 2577	. . . 4 |- U_f e. A dom f = U.{x | E.f e. A x = dom f}
28	22, 23, 27	3eqtr 1491	. . 3 |- dom F = U.{x | E.f e. A x = dom f}
29		ordeq 2945	. . 3 |- (dom F = U.{x | E.f e. A x = dom f} -> (Ord dom F <-> Ord U.{x | E.f e. A x = dom f}))
30	28, 29	ax-mp 7	. 2 |- (Ord dom F <-> Ord U.{x | E.f e. A x = dom f})
31	20, 30	mpbir 190	1 |- Ord dom F

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  U.cuni 2493  U_ciun 2556  Ord word 2937  Oncon0 2938  dom cdm 3160   |` cres 3162   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tfrlem10 3905  tfrlem13 3908

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-cnv 3176  df-dm 3178  df-rn 3179  df-fn 3183

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