Theorem tfrlem8 3924

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 1546	. . . . . . 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 1573	. . . . . . . . 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 1737	. . . . . . . . 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 1653	. . . . . . . 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 1546	. . . . . . . . . 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 3593	. . . . . . . . 9 |- (f Fn x -> dom f = x)
13	11, 12	sylan2 453	. . . . . . . 8 |- ((x e. On /\ f Fn x) -> dom f e. On)
14	13	19.23aiv 1297	. . . . . . 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 450	. . . . 5 |- ((f e. A /\ x = dom f) -> x e. On)
17	16	r19.23aiva 1747	. . . 4 |- (E.f e. A x = dom f -> x e. On)
18	17	abssi 2125	. . 3 |- {x | E.f e. A x = dom f} (_ On
19		ssorduni 2999	. . 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 3318	. . . 4 |- dom F = dom U. A
23		dmuni 3325	. . . 4 |- dom U. A = U_f e. A dom f
24		visset 1816	. . . . . 6 |- f e. V
25	24	dmex 3366	. . . . 5 |- dom f e. V
26	25	dfiun2 2591	. . . 4 |- U_f e. A dom f = U.{x | E.f e. A x = dom f}
27	22, 23, 26	3eqtr 1502	. . 3 |- dom F = U.{x | E.f e. A x = dom f}
28		ordeq 2961	. . 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}))
29	27, 28	ax-mp 7	. 2 |- (Ord dom F <-> Ord U.{x | E.f e. A x = dom f})
30	20, 29	mpbir 190	1 |- Ord dom F

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  A.wral 1648  E.wrex 1649   (_ wss 2050  U.cuni 2507  U_ciun 2570  Ord word 2953  Oncon0 2954  dom cdm 3176   |` cres 3178   Fn wfn 3183  ` cfv 3188

This theorem is referenced by:  tfrlem10 3926  tfrlem13 3929

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-cnv 3192  df-dm 3194  df-rn 3195  df-fn 3199

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