Theorem alephfp 8266

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 8267 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 8265	. 2 |- U. ( H " om ) e. ran aleph
3		isinfcard 8250	. . 3 |- ( ( om C_ U. ( H " om ) /\ ( card ` U. ( H " om ) ) = U. ( H " om ) ) <-> U. ( H " om ) e. ran aleph )
4		cardalephex 8248	. . . 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 481	. . 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 8246	. . . . . . . . 9 |- ( z e. On -> z C_ ( aleph ` z ) )
8		alephon 8227	. . . . . . . . . . 11 |- ( aleph ` z ) e. On
9	8	onirri 4812	. . . . . . . . . 10 |- -. ( aleph ` z ) e. ( aleph ` z )
10		frfnom 6876	. . . . . . . . . . . . . 14 |- ( rec ( aleph , om ) |` om ) Fn om
11	1	fneq1i 5493	. . . . . . . . . . . . . 14 |- ( H Fn om <-> ( rec ( aleph , om ) |` om ) Fn om )
12	10, 11	mpbir 209	. . . . . . . . . . . . 13 |- H Fn om
13		fnfun 5496	. . . . . . . . . . . . 13 |- ( H Fn om -> Fun H )
14		eluniima 5954	. . . . . . . . . . . . 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 8258	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
17	1	alephfplem3 8264	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( H ` v ) e. ran aleph )
18	16, 17	sseldi 3342	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( H ` v ) e. On )
19		alephord2i 8235	. . . . . . . . . . . . . . 15 |- ( ( H ` v ) e. On -> ( z e. ( H ` v ) -> ( aleph ` z ) e. ( aleph ` ( H ` v ) ) ) )
20	18, 19	syl 16	. . . . . . . . . . . . . 14 |- ( v e. om -> ( z e. ( H ` v ) -> ( aleph ` z ) e. ( aleph ` ( H ` v ) ) ) )
21	1	alephfplem2 8263	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) = ( aleph ` ( H ` v ) ) )
22		peano2 6485	. . . . . . . . . . . . . . . . . 18 |- ( v e. om -> suc v e. om )
23		fnfvelrn 5828	. . . . . . . . . . . . . . . . . . . 20 |- ( ( H Fn om /\ suc v e. om ) -> ( H ` suc v ) e. ran H )
24	12, 23	mpan 663	. . . . . . . . . . . . . . . . . . 19 |- ( suc v e. om -> ( H ` suc v ) e. ran H )
25		fnima 5517	. . . . . . . . . . . . . . . . . . . 20 |- ( H Fn om -> ( H " om ) = ran H )
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( H " om ) = ran H
27	24, 26	syl6eleqr 2524	. . . . . . . . . . . . . . . . . 18 |- ( suc v e. om -> ( H ` suc v ) e. ( H " om ) )
28	22, 27	syl 16	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) e. ( H " om ) )
29	21, 28	eqeltrrd 2508	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( aleph ` ( H ` v ) ) e. ( H " om ) )
30		elssuni 4109	. . . . . . . . . . . . . . . 16 |- ( ( aleph ` ( H ` v ) ) e. ( H " om ) -> ( aleph ` ( H ` v ) ) C_ U. ( H " om ) )
31	29, 30	syl 16	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( aleph ` ( H ` v ) ) C_ U. ( H " om ) )
32	31	sseld 3343	. . . . . . . . . . . . . 14 |- ( v e. om -> ( ( aleph ` z ) e. ( aleph ` ( H ` v ) ) -> ( aleph ` z ) e. U. ( H " om ) ) )
33	20, 32	syld 44	. . . . . . . . . . . . 13 |- ( v e. om -> ( z e. ( H ` v ) -> ( aleph ` z ) e. U. ( H " om ) ) )
34	33	rexlimiv 2825	. . . . . . . . . . . 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 2494	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. U. ( H " om ) <-> z e. ( aleph ` z ) ) )
37		eleq2 2494	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` z ) e. U. ( H " om ) <-> ( aleph ` z ) e. ( aleph ` z ) ) )
38	36, 37	imbi12d 320	. . . . . . . . . . 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 561	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) )
42		eloni 4716	. . . . . . . . . 10 |- ( z e. On -> Ord z )
43	8	onordi 4810	. . . . . . . . . 10 |- Ord ( aleph ` z )
44		ordtri4 4743	. . . . . . . . . 10 |- ( ( Ord z /\ Ord ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
45	42, 43, 44	sylancl 655	. . . . . . . . 9 |- ( z e. On -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
46	45	adantr 462	. . . . . . . 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 2442	. . . . . . . 8 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
49	48	adantl 463	. . . . . . 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 2438	. . . . 5 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> U. ( H " om ) = z )
52	51	fveq2d 5683	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) )
53		eqeq2 2442	. . . . 5 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` U. ( H " om ) ) = U. ( H " om ) <-> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) ) )
54	53	adantl 463	. . . 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 2826	. 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 369   = wceq 1362   e. wcel 1755   E.wrex 2706   C_ wss 3316   U.cuni 4079   Ord word 4705   Oncon0 4706   suc csuc 4708   ran crn 4828   |` cres 4829   "cima 4830   Fun wfun 5400   Fn wfn 5401   ` cfv 5406   omcom 6465   reccrdg 6851   cardccrd 8093   alephcale 8094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-oi 7712  df-har 7761  df-card 8097  df-aleph 8098

This theorem is referenced by:  alephfp2  8267