Theorem alephfp 6048

Description: The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 6049 for an abbreviated version just showing existence.

Hypothesis

Ref	Expression
alephfplem.1	|- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)

Assertion

Ref	Expression
alephfp	|- (aleph` U.(H"om)) = U.(H"om)

Distinct variable group:   x,y

Proof of Theorem alephfp

Step	Hyp	Ref	Expression
1		alephfplem.1	. . . 4 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
2	1	alephfplem4 6047	. . 3 |- U.(H"om) e. ran aleph
3		isinfcard 6035	. . . 4 |- ((om C_ U.(H"om) /\ (card` U.(H"om)) = U.(H"om)) <-> U.(H"om) e. ran aleph)
4		cardalephex 6034	. . . . 5 |- (om C_ U.(H"om) -> ((card` U.(H"om)) = U.(H"om) <-> E.z e. On U.(H"om) = (aleph` z)))
5	4	biimpa 460	. . . 4 |- ((om C_ U.(H"om) /\ (card` U.(H"om)) = U.(H"om)) -> E.z e. On U.(H"om) = (aleph` z))
6	3, 5	sylbir 218	. . 3 |- (U.(H"om) e. ran aleph -> E.z e. On U.(H"om) = (aleph` z))
7	2, 6	ax-mp 7	. 2 |- E.z e. On U.(H"om) = (aleph` z)
8		alephle 6032	. . . . . . . . 9 |- (z e. On -> z C_ (aleph` z))
9		elirr 5701	. . . . . . . . . 10 |- -. (aleph` z) e. (aleph` z)
10		frfnom 5159	. . . . . . . . . . . . . . 15 |- (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om
11	1	fneq1i 4507	. . . . . . . . . . . . . . 15 |- (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om)
12	10, 11	mpbir 207	. . . . . . . . . . . . . 14 |- H Fn om
13		fnfun 4510	. . . . . . . . . . . . . 14 |- (H Fn om -> Fun H)
14	12, 13	ax-mp 7	. . . . . . . . . . . . 13 |- Fun H
15		eluniima 4843	. . . . . . . . . . . . 13 |- (Fun H -> (z e. U.(H"om) <-> E.v e. om z e. (H` v)))
16	14, 15	ax-mp 7	. . . . . . . . . . . 12 |- (z e. U.(H"om) <-> E.v e. om z e. (H` v))
17	1	alephfplem3 6046	. . . . . . . . . . . . . . 15 |- (v e. om -> (H` v) e. ran aleph)
18		alephsson 6042	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
19	18	sseli 2617	. . . . . . . . . . . . . . 15 |- ((H` v) e. ran aleph -> (H` v) e. On)
20		alephord2i 6025	. . . . . . . . . . . . . . 15 |- ((H` v) e. On -> (z e. (H` v) -> (aleph` z) e. (aleph` (H` v))))
21	17, 19, 20	3syl 24	. . . . . . . . . . . . . 14 |- (v e. om -> (z e. (H` v) -> (aleph` z) e. (aleph` (H` v))))
22	1	alephfplem2 6045	. . . . . . . . . . . . . . . . 17 |- (v e. om -> (H` suc v) = (aleph` (H` v)))
23		peano2 3972	. . . . . . . . . . . . . . . . . 18 |- (v e. om -> suc v e. om)
24		fnfvelrn 4786	. . . . . . . . . . . . . . . . . . . 20 |- ((H Fn om /\ suc v e. om) -> (H` suc v) e. ran H)
25	12, 24	mpan 759	. . . . . . . . . . . . . . . . . . 19 |- (suc v e. om -> (H` suc v) e. ran H)
26		fnima 4530	. . . . . . . . . . . . . . . . . . . 20 |- (H Fn om -> (H"om) = ran H)
27	12, 26	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- (H"om) = ran H
28	25, 27	syl6eleqr 1982	. . . . . . . . . . . . . . . . . 18 |- (suc v e. om -> (H` suc v) e. (H"om))
29	23, 28	syl 12	. . . . . . . . . . . . . . . . 17 |- (v e. om -> (H` suc v) e. (H"om))
30	22, 29	eqeltrrd 1972	. . . . . . . . . . . . . . . 16 |- (v e. om -> (aleph` (H` v)) e. (H"om))
31		elssuni 3206	. . . . . . . . . . . . . . . 16 |- ((aleph` (H` v)) e. (H"om) -> (aleph` (H` v)) C_ U.(H"om))
32	30, 31	syl 12	. . . . . . . . . . . . . . 15 |- (v e. om -> (aleph` (H` v)) C_ U.(H"om))
33	32	sseld 2619	. . . . . . . . . . . . . 14 |- (v e. om -> ((aleph` z) e. (aleph` (H` v)) -> (aleph` z) e. U.(H"om)))
34	21, 33	syld 30	. . . . . . . . . . . . 13 |- (v e. om -> (z e. (H` v) -> (aleph` z) e. U.(H"om)))
35	34	r19.23aiv 2211	. . . . . . . . . . . 12 |- (E.v e. om z e. (H` v) -> (aleph` z) e. U.(H"om))
36	16, 35	sylbi 216	. . . . . . . . . . 11 |- (z e. U.(H"om) -> (aleph` z) e. U.(H"om))
37		eleq2 1958	. . . . . . . . . . . 12 |- (U.(H"om) = (aleph` z) -> (z e. U.(H"om) <-> z e. (aleph` z)))
38		eleq2 1958	. . . . . . . . . . . 12 |- (U.(H"om) = (aleph` z) -> ((aleph` z) e. U.(H"om) <-> (aleph` z) e. (aleph` z)))
39	37, 38	imbi12d 688	. . . . . . . . . . 11 |- (U.(H"om) = (aleph` z) -> ((z e. U.(H"om) -> (aleph` z) e. U.(H"om)) <-> (z e. (aleph` z) -> (aleph` z) e. (aleph` z))))
40	36, 39	mpbii 210	. . . . . . . . . 10 |- (U.(H"om) = (aleph` z) -> (z e. (aleph` z) -> (aleph` z) e. (aleph` z)))
41	9, 40	mtoi 122	. . . . . . . . 9 |- (U.(H"om) = (aleph` z) -> -. z e. (aleph` z))
42	8, 41	anim12i 360	. . . . . . . 8 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z C_ (aleph` z) /\ -. z e. (aleph` z)))
43		ordtri4 3699	. . . . . . . . . 10 |- ((Ord z /\ Ord (aleph` z)) -> (z = (aleph` z) <-> (z C_ (aleph` z) /\ -. z e. (aleph` z))))
44		eloni 3667	. . . . . . . . . 10 |- (z e. On -> Ord z)
45		alephon 5876	. . . . . . . . . . 11 |- (aleph` z) e. On
46	45	onordi 3774	. . . . . . . . . 10 |- Ord (aleph` z)
47	43, 44, 46	sylancl 525	. . . . . . . . 9 |- (z e. On -> (z = (aleph` z) <-> (z C_ (aleph` z) /\ -. z e. (aleph` z))))
48	47	adantr 425	. . . . . . . 8 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z = (aleph` z) <-> (z C_ (aleph` z) /\ -. z e. (aleph` z))))
49	42, 48	mpbird 213	. . . . . . 7 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> z = (aleph` z))
50		eqeq2 1893	. . . . . . . 8 |- (U.(H"om) = (aleph` z) -> (z = U.(H"om) <-> z = (aleph` z)))
51	50	adantl 424	. . . . . . 7 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z = U.(H"om) <-> z = (aleph` z)))
52	49, 51	mpbird 213	. . . . . 6 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> z = U.(H"om))
53	52	eqcomd 1889	. . . . 5 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> U.(H"om) = z)
54	53	fveq2d 4685	. . . 4 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (aleph` U.(H"om)) = (aleph` z))
55		eqeq2 1893	. . . . 5 |- (U.(H"om) = (aleph` z) -> ((aleph` U.(H"om)) = U.(H"om) <-> (aleph` U.(H"om)) = (aleph` z)))
56	55	adantl 424	. . . 4 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> ((aleph` U.(H"om)) = U.(H"om) <-> (aleph` U.(H"om)) = (aleph` z)))
57	54, 56	mpbird 213	. . 3 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (aleph` U.(H"om)) = U.(H"om))
58	57	r19.23aiva 2212	. 2 |- (E.z e. On U.(H"om) = (aleph` z) -> (aleph` U.(H"om)) = U.(H"om))
59	7, 58	ax-mp 7	1 |- (aleph` U.(H"om)) = U.(H"om)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  (/)c0 2875  U.cuni 3177  {copab 3395  Ord word 3656  Oncon0 3657  suc csuc 3659  omcom 3949  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  reccrdg 5139  cardccrd 5859  alephcale 5860

This theorem is referenced by:  alephfp2 6049

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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  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-lim 3662  df-suc 3663  df-om 3950  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862  df-aleph 5863

Copyright terms: Public domain