Theorem alephfp 4880

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 4881 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 4879	. . 3 |- U.(H"om) e. ran aleph
3		isinfcard 4867	. . . 4 |- ((om (_ U.(H"om) /\ (card` U.(H"om)) = U.(H"om)) <-> U.(H"om) e. ran aleph)
4		cardalephex 4866	. . . . 5 |- (om (_ U.(H"om) -> ((card` U.(H"om)) = U.(H"om) <-> E.z e. On U.(H"om) = (aleph` z)))
5	4	biimpa 416	. . . 4 |- ((om (_ U.(H"om) /\ (card` U.(H"om)) = U.(H"om)) -> E.z e. On U.(H"om) = (aleph` z))
6	3, 5	sylbir 201	. . 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 4864	. . . . . . . . 9 |- (z e. On -> z (_ (aleph` z))
9		elirr 4579	. . . . . . . . . 10 |- -. (aleph` z) e. (aleph` z)
10		frfnom 3942	. . . . . . . . . . . . . . 15 |- (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om
11		fneq1 3574	. . . . . . . . . . . . . . . 16 |- (H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) -> (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om))
12	1, 11	ax-mp 7	. . . . . . . . . . . . . . 15 |- (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om)
13	10, 12	mpbir 190	. . . . . . . . . . . . . 14 |- H Fn om
14		fnfun 3577	. . . . . . . . . . . . . 14 |- (H Fn om -> Fun H)
15	13, 14	ax-mp 7	. . . . . . . . . . . . 13 |- Fun H
16		eluniima 3858	. . . . . . . . . . . . 13 |- (Fun H -> (z e. U.(H"om) <-> E.v e. om z e. (H` v)))
17	15, 16	ax-mp 7	. . . . . . . . . . . 12 |- (z e. U.(H"om) <-> E.v e. om z e. (H` v))
18	1	alephfplem3 4878	. . . . . . . . . . . . . . 15 |- (v e. om -> (H` v) e. ran aleph)
19		alephsson 4874	. . . . . . . . . . . . . . . 16 |- ran aleph (_ On
20	19	sseli 2061	. . . . . . . . . . . . . . 15 |- ((H` v) e. ran aleph -> (H` v) e. On)
21		alephord2i 4857	. . . . . . . . . . . . . . 15 |- ((H` v) e. On -> (z e. (H` v) -> (aleph` z) e. (aleph` (H` v))))
22	18, 20, 21	3syl 20	. . . . . . . . . . . . . 14 |- (v e. om -> (z e. (H` v) -> (aleph` z) e. (aleph` (H` v))))
23	1	alephfplem2 4877	. . . . . . . . . . . . . . . . 17 |- (v e. om -> (H` suc v) = (aleph` (H` v)))
24		peano2 3145	. . . . . . . . . . . . . . . . . 18 |- (v e. om -> suc v e. om)
25		fnfvelrn 3804	. . . . . . . . . . . . . . . . . . . 20 |- ((H Fn om /\ suc v e. om) -> (H` suc v) e. ran H)
26	13, 25	mpan 694	. . . . . . . . . . . . . . . . . . 19 |- (suc v e. om -> (H` suc v) e. ran H)
27		fnima 3596	. . . . . . . . . . . . . . . . . . . 20 |- (H Fn om -> (H"om) = ran H)
28	13, 27	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- (H"om) = ran H
29	26, 28	syl6eleqr 1556	. . . . . . . . . . . . . . . . . 18 |- (suc v e. om -> (H` suc v) e. (H"om))
30	24, 29	syl 10	. . . . . . . . . . . . . . . . 17 |- (v e. om -> (H` suc v) e. (H"om))
31	23, 30	eqeltrrd 1546	. . . . . . . . . . . . . . . 16 |- (v e. om -> (aleph` (H` v)) e. (H"om))
32		elssuni 2521	. . . . . . . . . . . . . . . 16 |- ((aleph` (H` v)) e. (H"om) -> (aleph` (H` v)) (_ U.(H"om))
33	31, 32	syl 10	. . . . . . . . . . . . . . 15 |- (v e. om -> (aleph` (H` v)) (_ U.(H"om))
34	33	sseld 2063	. . . . . . . . . . . . . 14 |- (v e. om -> ((aleph` z) e. (aleph` (H` v)) -> (aleph` z) e. U.(H"om)))
35	22, 34	syld 27	. . . . . . . . . . . . 13 |- (v e. om -> (z e. (H` v) -> (aleph` z) e. U.(H"om)))
36	35	r19.23aiv 1740	. . . . . . . . . . . 12 |- (E.v e. om z e. (H` v) -> (aleph` z) e. U.(H"om))
37	17, 36	sylbi 199	. . . . . . . . . . 11 |- (z e. U.(H"om) -> (aleph` z) e. U.(H"om))
38		eleq2 1532	. . . . . . . . . . . 12 |- (U.(H"om) = (aleph` z) -> (z e. U.(H"om) <-> z e. (aleph` z)))
39		eleq2 1532	. . . . . . . . . . . 12 |- (U.(H"om) = (aleph` z) -> ((aleph` z) e. U.(H"om) <-> (aleph` z) e. (aleph` z)))
40	38, 39	imbi12d 625	. . . . . . . . . . 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))))
41	37, 40	mpbii 193	. . . . . . . . . 10 |- (U.(H"om) = (aleph` z) -> (z e. (aleph` z) -> (aleph` z) e. (aleph` z)))
42	9, 41	mtoi 107	. . . . . . . . 9 |- (U.(H"om) = (aleph` z) -> -. z e. (aleph` z))
43	8, 42	anim12i 333	. . . . . . . 8 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z (_ (aleph` z) /\ -. z e. (aleph` z)))
44		eloni 2953	. . . . . . . . . 10 |- (z e. On -> Ord z)
45		alephon 4845	. . . . . . . . . . . 12 |- (aleph` z) e. On
46	45	onord 3090	. . . . . . . . . . 11 |- Ord (aleph` z)
47		ordtri4 2979	. . . . . . . . . . 11 |- ((Ord z /\ Ord (aleph` z)) -> (z = (aleph` z) <-> (z (_ (aleph` z) /\ -. z e. (aleph` z))))
48	46, 47	mpan2 695	. . . . . . . . . 10 |- (Ord z -> (z = (aleph` z) <-> (z (_ (aleph` z) /\ -. z e. (aleph` z))))
49	44, 48	syl 10	. . . . . . . . 9 |- (z e. On -> (z = (aleph` z) <-> (z (_ (aleph` z) /\ -. z e. (aleph` z))))
50	49	adantr 389	. . . . . . . 8 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z = (aleph` z) <-> (z (_ (aleph` z) /\ -. z e. (aleph` z))))
51	43, 50	mpbird 196	. . . . . . 7 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> z = (aleph` z))
52		eqeq2 1481	. . . . . . . 8 |- (U.(H"om) = (aleph` z) -> (z = U.(H"om) <-> z = (aleph` z)))
53	52	adantl 388	. . . . . . 7 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (z = U.(H"om) <-> z = (aleph` z)))
54	51, 53	mpbird 196	. . . . . 6 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> z = U.(H"om))
55	54	eqcomd 1477	. . . . 5 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> U.(H"om) = z)
56	55	fveq2d 3719	. . . 4 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (aleph` U.(H"om)) = (aleph` z))
57		eqeq2 1481	. . . . 5 |- (U.(H"om) = (aleph` z) -> ((aleph` U.(H"om)) = U.(H"om) <-> (aleph` U.(H"om)) = (aleph` z)))
58	57	adantl 388	. . . 4 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> ((aleph` U.(H"om)) = U.(H"om) <-> (aleph` U.(H"om)) = (aleph` z)))
59	56, 58	mpbird 196	. . 3 |- ((z e. On /\ U.(H"om) = (aleph` z)) -> (aleph` U.(H"om)) = U.(H"om))
60	59	r19.23aiva 1741	. 2 |- (E.z e. On U.(H"om) = (aleph` z) -> (aleph` U.(H"om)) = U.(H"om))
61	7, 60	ax-mp 7	1 |- (aleph` U.(H"om)) = U.(H"om)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wrex 1643   (_ wss 2043  (/)c0 2276  U.cuni 2498  {copab 2661  Ord word 2942  Oncon0 2943  suc csuc 2945  omcom 3126  ran crn 3166   |` cres 3167  "cima 3168  Fun wfun 3171   Fn wfn 3172  ` cfv 3177  reccrdg 3922  cardccrd 4793  alephcale 4794

This theorem is referenced by:  alephfp2 4881

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  ax-reg 4573  ax-inf2 4605  ax-ac 4724

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-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529