Theorem alephfp 8379

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 8380 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 8378	. 2 |- U. ( H " om ) e. ran aleph
3		isinfcard 8363	. . 3 |- ( ( om C_ U. ( H " om ) /\ ( card ` U. ( H " om ) ) = U. ( H " om ) ) <-> U. ( H " om ) e. ran aleph )
4		cardalephex 8361	. . . 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 484	. . 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 8359	. . . . . . . . 9 |- ( z e. On -> z C_ ( aleph ` z ) )
8		alephon 8340	. . . . . . . . . . 11 |- ( aleph ` z ) e. On
9	8	onirri 4923	. . . . . . . . . 10 |- -. ( aleph ` z ) e. ( aleph ` z )
10		frfnom 6990	. . . . . . . . . . . . . 14 |- ( rec ( aleph , om ) |` om ) Fn om
11	1	fneq1i 5603	. . . . . . . . . . . . . 14 |- ( H Fn om <-> ( rec ( aleph , om ) |` om ) Fn om )
12	10, 11	mpbir 209	. . . . . . . . . . . . 13 |- H Fn om
13		fnfun 5606	. . . . . . . . . . . . 13 |- ( H Fn om -> Fun H )
14		eluniima 6066	. . . . . . . . . . . . 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 8371	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
17	1	alephfplem3 8377	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( H ` v ) e. ran aleph )
18	16, 17	sseldi 3452	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( H ` v ) e. On )
19		alephord2i 8348	. . . . . . . . . . . . . . 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 8376	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) = ( aleph ` ( H ` v ) ) )
22		peano2 6596	. . . . . . . . . . . . . . . . . 18 |- ( v e. om -> suc v e. om )
23		fnfvelrn 5939	. . . . . . . . . . . . . . . . . . . 20 |- ( ( H Fn om /\ suc v e. om ) -> ( H ` suc v ) e. ran H )
24	12, 23	mpan 670	. . . . . . . . . . . . . . . . . . 19 |- ( suc v e. om -> ( H ` suc v ) e. ran H )
25		fnima 5627	. . . . . . . . . . . . . . . . . . . 20 |- ( H Fn om -> ( H " om ) = ran H )
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( H " om ) = ran H
27	24, 26	syl6eleqr 2550	. . . . . . . . . . . . . . . . . 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 2540	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( aleph ` ( H ` v ) ) e. ( H " om ) )
30		elssuni 4219	. . . . . . . . . . . . . . . 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 3453	. . . . . . . . . . . . . 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 2931	. . . . . . . . . . . 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 2524	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. U. ( H " om ) <-> z e. ( aleph ` z ) ) )
37		eleq2 2524	. . . . . . . . . . . 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 566	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) )
42		eloni 4827	. . . . . . . . . 10 |- ( z e. On -> Ord z )
43	8	onordi 4921	. . . . . . . . . 10 |- Ord ( aleph ` z )
44		ordtri4 4854	. . . . . . . . . 10 |- ( ( Ord z /\ Ord ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
45	42, 43, 44	sylancl 662	. . . . . . . . 9 |- ( z e. On -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
46	45	adantr 465	. . . . . . . 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 2466	. . . . . . . 8 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
49	48	adantl 466	. . . . . . 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 2459	. . . . 5 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> U. ( H " om ) = z )
52	51	fveq2d 5793	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) )
53		eqeq2 2466	. . . . 5 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` U. ( H " om ) ) = U. ( H " om ) <-> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) ) )
54	53	adantl 466	. . . 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 2932	. 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 1370   e. wcel 1758   E.wrex 2796   C_ wss 3426   U.cuni 4189   Ord word 4816   Oncon0 4817   suc csuc 4819   ran crn 4939   |` cres 4940   "cima 4941   Fun wfun 5510   Fn wfn 5511   ` cfv 5516   omcom 6576   reccrdg 6965   cardccrd 8206   alephcale 8207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-om 6577  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-oi 7825  df-har 7874  df-card 8210  df-aleph 8211

This theorem is referenced by:  alephfp2  8380