Theorem tfrlem13 5131

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 5126	. . 3 |- Ord dom F
4		ordirr 3676	. . . 4 |- (Ord dom F -> -. dom F e. dom F)
5		tfrlem.3	. . . . . 6 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
6	1, 2, 5	tfrlem12 5130	. . . . 5 |- (dom F e. On -> C e. A)
7		sucidg 3743	. . . . . 6 |- (dom F e. On -> dom F e. suc dom F)
8	1, 2, 5	tfrlem10 5128	. . . . . . 7 |- (dom F e. On -> C Fn suc dom F)
9		fndm 4512	. . . . . . 7 |- (C Fn suc dom F -> dom C = suc dom F)
10	8, 9	syl 12	. . . . . 6 |- (dom F e. On -> dom C = suc dom F)
11	7, 10	eleqtrrd 1974	. . . . 5 |- (dom F e. On -> dom F e. dom C)
12		elssuni 3206	. . . . . . 7 |- (C e. A -> C C_ U.A)
13	12, 2	syl6ssr 2664	. . . . . 6 |- (C e. A -> C C_ F)
14		dmss 4156	. . . . . 6 |- (C C_ F -> dom C C_ dom F)
15		ssel 2615	. . . . . 6 |- (dom C C_ dom F -> (dom F e. dom C -> dom F e. dom F))
16	13, 14, 15	3syl 24	. . . . 5 |- (C e. A -> (dom F e. dom C -> dom F e. dom F))
17	6, 11, 16	sylc 83	. . . 4 |- (dom F e. On -> dom F e. dom F)
18	4, 17	nsyl 131	. . 3 |- (Ord dom F -> -. dom F e. On)
19	3, 18	ax-mp 7	. 2 |- -. dom F e. On
20		ordeleqon 3866	. . 3 |- (Ord dom F <-> (dom F e. On \/ dom F = On))
21	3, 20	mpbi 206	. 2 |- (dom F e. On \/ dom F = On)
22		orel1 271	. 2 |- (-. dom F e. On -> ((dom F e. On \/ dom F = On) -> dom F = On))
23	19, 21, 22	mp2 54	1 |- dom F = On

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   u. cun 2591   C_ wss 2593  {csn 3044  <.cop 3046  U.cuni 3177  Ord word 3656  Oncon0 3657  suc csuc 3659  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  tfr1 5132  tfr2 5133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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