Theorem tfrlem13 3914

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 3909	. . 3 |- Ord dom F
4		ordirr 2961	. . . 4 |- (Ord dom F -> -. dom F e. dom F)
5		elssuni 2521	. . . . . . 7 |- (C e. A -> C (_ U.A)
6	5, 2	syl6ssr 2104	. . . . . 6 |- (C e. A -> C (_ F)
7		dmss 3305	. . . . . 6 |- (C (_ F -> dom C (_ dom F)
8		ssel 2059	. . . . . 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 3913	. . . . 5 |- (dom F e. On -> C e. A)
12		sucidg 3047	. . . . . 6 |- (dom F e. On -> dom F e. suc dom F)
13	1, 2, 10	tfrlem10 3911	. . . . . . 7 |- (dom F e. On -> C Fn suc dom F)
14		fndm 3579	. . . . . . 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 1548	. . . . 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 2985	. . 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 954   e. wcel 956  {cab 1461  A.wral 1642  E.wrex 1643   u. cun 2041   (_ wss 2043  {csn 2405  <.cop 2407  U.cuni 2498  Ord word 2942  Oncon0 2943  suc csuc 2945  dom cdm 3165   |` cres 3167   Fn wfn 3172  ` cfv 3177

This theorem is referenced by:  tfr1 3915  tfr2 3916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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