Theorem alephfp 8521

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 8522 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
alephfplem.1	|- H = ( rec ( aleph , om ) |` om )

Assertion

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

Proof of Theorem alephfp

Dummy variables z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfplem.1	. . 3 |- H = ( rec ( aleph , om ) |` om )
2	1	alephfplem4 8520	. 2 |- U. ( H " om ) e. ran aleph
3		isinfcard 8505	. . 3 |- ( ( om C_ U. ( H " om ) /\ ( card ` U. ( H " om ) ) = U. ( H " om ) ) <-> U. ( H " om ) e. ran aleph )
4		cardalephex 8503	. . . 4 |- ( om C_ U. ( H " om ) -> ( ( card ` U. ( H " om ) ) = U. ( H " om ) <-> E. z e. On U. ( H " om ) = ( aleph ` z ) ) )
5	4	biimpa 482	. . 3 |- ( ( 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 213	. 2 |- ( U. ( H " om ) e. ran aleph -> E. z e. On U. ( H " om ) = ( aleph ` z ) )
7		alephle 8501	. . . . . . . . 9 |- ( z e. On -> z C_ ( aleph ` z ) )
8		alephon 8482	. . . . . . . . . . 11 |- ( aleph ` z ) e. On
9	8	onirri 5516	. . . . . . . . . 10 |- -. ( aleph ` z ) e. ( aleph ` z )
10		frfnom 7137	. . . . . . . . . . . . . 14 |- ( rec ( aleph , om ) |` om ) Fn om
11	1	fneq1i 5656	. . . . . . . . . . . . . 14 |- ( H Fn om <-> ( rec ( aleph , om ) |` om ) Fn om )
12	10, 11	mpbir 209	. . . . . . . . . . . . 13 |- H Fn om
13		fnfun 5659	. . . . . . . . . . . . 13 |- ( H Fn om -> Fun H )
14		eluniima 6143	. . . . . . . . . . . . 13 |- ( Fun H -> ( z e. U. ( H " om ) <-> E. v e. om z e. ( H ` v ) ) )
15	12, 13, 14	mp2b 10	. . . . . . . . . . . 12 |- ( z e. U. ( H " om ) <-> E. v e. om z e. ( H ` v ) )
16		alephsson 8513	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
17	1	alephfplem3 8519	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( H ` v ) e. ran aleph )
18	16, 17	sseldi 3440	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( H ` v ) e. On )
19		alephord2i 8490	. . . . . . . . . . . . . . 15 |- ( ( H ` v ) e. On -> ( z e. ( H ` v ) -> ( aleph ` z ) e. ( aleph ` ( H ` v ) ) ) )
20	18, 19	syl 17	. . . . . . . . . . . . . 14 |- ( v e. om -> ( z e. ( H ` v ) -> ( aleph ` z ) e. ( aleph ` ( H ` v ) ) ) )
21	1	alephfplem2 8518	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) = ( aleph ` ( H ` v ) ) )
22		peano2 6704	. . . . . . . . . . . . . . . . . 18 |- ( v e. om -> suc v e. om )
23		fnfvelrn 6006	. . . . . . . . . . . . . . . . . . . 20 |- ( ( H Fn om /\ suc v e. om ) -> ( H ` suc v ) e. ran H )
24	12, 23	mpan 668	. . . . . . . . . . . . . . . . . . 19 |- ( suc v e. om -> ( H ` suc v ) e. ran H )
25		fnima 5680	. . . . . . . . . . . . . . . . . . . 20 |- ( H Fn om -> ( H " om ) = ran H )
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( H " om ) = ran H
27	24, 26	syl6eleqr 2501	. . . . . . . . . . . . . . . . . 18 |- ( suc v e. om -> ( H ` suc v ) e. ( H " om ) )
28	22, 27	syl 17	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) e. ( H " om ) )
29	21, 28	eqeltrrd 2491	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( aleph ` ( H ` v ) ) e. ( H " om ) )
30		elssuni 4220	. . . . . . . . . . . . . . . 16 |- ( ( aleph ` ( H ` v ) ) e. ( H " om ) -> ( aleph ` ( H ` v ) ) C_ U. ( H " om ) )
31	29, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( aleph ` ( H ` v ) ) C_ U. ( H " om ) )
32	31	sseld 3441	. . . . . . . . . . . . . 14 |- ( v e. om -> ( ( aleph ` z ) e. ( aleph ` ( H ` v ) ) -> ( aleph ` z ) e. U. ( H " om ) ) )
33	20, 32	syld 42	. . . . . . . . . . . . 13 |- ( v e. om -> ( z e. ( H ` v ) -> ( aleph ` z ) e. U. ( H " om ) ) )
34	33	rexlimiv 2890	. . . . . . . . . . . 12 |- ( E. v e. om z e. ( H ` v ) -> ( aleph ` z ) e. U. ( H " om ) )
35	15, 34	sylbi 195	. . . . . . . . . . 11 |- ( z e. U. ( H " om ) -> ( aleph ` z ) e. U. ( H " om ) )
36		eleq2 2475	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. U. ( H " om ) <-> z e. ( aleph ` z ) ) )
37		eleq2 2475	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` z ) e. U. ( H " om ) <-> ( aleph ` z ) e. ( aleph ` z ) ) )
38	36, 37	imbi12d 318	. . . . . . . . . . 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 ) ) ) )
39	35, 38	mpbii 211	. . . . . . . . . 10 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. ( aleph ` z ) -> ( aleph ` z ) e. ( aleph ` z ) ) )
40	9, 39	mtoi 178	. . . . . . . . 9 |- ( U. ( H " om ) = ( aleph ` z ) -> -. z e. ( aleph ` z ) )
41	7, 40	anim12i 564	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) )
42		eloni 5420	. . . . . . . . . 10 |- ( z e. On -> Ord z )
43	8	onordi 5514	. . . . . . . . . 10 |- Ord ( aleph ` z )
44		ordtri4 5447	. . . . . . . . . 10 |- ( ( Ord z /\ Ord ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
45	42, 43, 44	sylancl 660	. . . . . . . . 9 |- ( z e. On -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
46	45	adantr 463	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
47	41, 46	mpbird 232	. . . . . . 7 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> z = ( aleph ` z ) )
48		eqeq2 2417	. . . . . . . 8 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
49	48	adantl 464	. . . . . . 7 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
50	47, 49	mpbird 232	. . . . . 6 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> z = U. ( H " om ) )
51	50	eqcomd 2410	. . . . 5 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> U. ( H " om ) = z )
52	51	fveq2d 5853	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) )
53		eqeq2 2417	. . . . 5 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` U. ( H " om ) ) = U. ( H " om ) <-> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) ) )
54	53	adantl 464	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( ( aleph ` U. ( H " om ) ) = U. ( H " om ) <-> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) ) )
55	52, 54	mpbird 232	. . 3 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = U. ( H " om ) )
56	55	rexlimiva 2892	. 2 |- ( E. z e. On U. ( H " om ) = ( aleph ` z ) -> ( aleph ` U. ( H " om ) ) = U. ( H " om ) )
57	2, 6, 56	mp2b 10	1 |- ( aleph ` U. ( H " om ) ) = U. ( H " om )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   e. wcel 1842   E.wrex 2755   C_ wss 3414   U.cuni 4191   ran crn 4824   |` cres 4825   "cima 4826   Ord word 5409   Oncon0 5410   suc csuc 5412   Fun wfun 5563   Fn wfn 5564   ` cfv 5569   omcom 6683   reccrdg 7112   cardccrd 8348   alephcale 8349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-har 8018  df-card 8352  df-aleph 8353

This theorem is referenced by:  alephfp2  8522