Theorem tfrlem13 3908

Description: Lemma for transfinite recursion. If dom F is in On, then C is acceptable, and thus a subset of F, but dom C is bigger than dom F. This is a contradiction, so dom F must be On.

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
tfrlem.3	|- C = (F u. {<.dom F, (G` (F |` dom F))>.})

Assertion

Ref	Expression
tfrlem13	|- dom F = On

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

Proof of Theorem tfrlem13

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
2		tfrlem.2	. . . 4 |- F = U.A
3	1, 2	tfrlem8 3903	. . 3 |- Ord dom F
4		ordirr 2956	. . . 4 |- (Ord dom F -> -. dom F e. dom F)
5		elssuni 2516	. . . . . . 7 |- (C e. A -> C (_ U.A)
6	5, 2	syl6ssr 2098	. . . . . 6 |- (C e. A -> C (_ F)
7		dmss 3299	. . . . . 6 |- (C (_ F -> dom C (_ dom F)
8		ssel 2053	. . . . . 6 |- (dom C (_ dom F -> (dom F e. dom C -> dom F e. dom F))
9	6, 7, 8	3syl 20	. . . . 5 |- (C e. A -> (dom F e. dom C -> dom F e. dom F))
10		tfrlem.3	. . . . . 6 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
11	1, 2, 10	tfrlem12 3907	. . . . 5 |- (dom F e. On -> C e. A)
12		sucidg 3042	. . . . . 6 |- (dom F e. On -> dom F e. suc dom F)
13	1, 2, 10	tfrlem10 3905	. . . . . . 7 |- (dom F e. On -> C Fn suc dom F)
14		fndm 3573	. . . . . . 7 |- (C Fn suc dom F -> dom C = suc dom F)
15	13, 14	syl 10	. . . . . 6 |- (dom F e. On -> dom C = suc dom F)
16	12, 15	eleqtrrd 1543	. . . . 5 |- (dom F e. On -> dom F e. dom C)
17	9, 11, 16	sylc 68	. . . 4 |- (dom F e. On -> dom F e. dom F)
18	4, 17	nsyl 116	. . 3 |- (Ord dom F -> -. dom F e. On)
19	3, 18	ax-mp 7	. 2 |- -. dom F e. On
20		ordeleqon 2980	. . 3 |- (Ord dom F <-> (dom F e. On \/ dom F = On))
21	3, 20	mpbi 189	. 2 |- (dom F e. On \/ dom F = On)
22		orel1 251	. 2 |- (-. dom F e. On -> ((dom F e. On \/ dom F = On) -> dom F = On))
23	19, 21, 22	mp2 43	1 |- dom F = On

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637  E.wrex 1638   u. cun 2035   (_ wss 2037  {csn 2399  <.cop 2401  U.cuni 2493  Ord word 2937  Oncon0 2938  suc csuc 2940  dom cdm 3160   |` cres 3162   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tfr1 3909  tfr2 3910

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-9 962  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-rep 2683  ax-sep 2693  ax-nul 2700  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-rab 1644  df-v 1803  df-sbc 1932  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-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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