Theorem alephfp 8478

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 8479 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 8477	. 2 |- U. ( H " om ) e. ran aleph
3		isinfcard 8462	. . 3 |- ( ( om C_ U. ( H " om ) /\ ( card ` U. ( H " om ) ) = U. ( H " om ) ) <-> U. ( H " om ) e. ran aleph )
4		cardalephex 8460	. . . 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 8458	. . . . . . . . 9 |- ( z e. On -> z C_ ( aleph ` z ) )
8		alephon 8439	. . . . . . . . . . 11 |- ( aleph ` z ) e. On
9	8	onirri 4977	. . . . . . . . . 10 |- -. ( aleph ` z ) e. ( aleph ` z )
10		frfnom 7090	. . . . . . . . . . . . . 14 |- ( rec ( aleph , om ) |` om ) Fn om
11	1	fneq1i 5666	. . . . . . . . . . . . . 14 |- ( H Fn om <-> ( rec ( aleph , om ) |` om ) Fn om )
12	10, 11	mpbir 209	. . . . . . . . . . . . 13 |- H Fn om
13		fnfun 5669	. . . . . . . . . . . . 13 |- ( H Fn om -> Fun H )
14		eluniima 6141	. . . . . . . . . . . . 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 8470	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
17	1	alephfplem3 8476	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( H ` v ) e. ran aleph )
18	16, 17	sseldi 3495	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( H ` v ) e. On )
19		alephord2i 8447	. . . . . . . . . . . . . . 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 8475	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) = ( aleph ` ( H ` v ) ) )
22		peano2 6691	. . . . . . . . . . . . . . . . . 18 |- ( v e. om -> suc v e. om )
23		fnfvelrn 6009	. . . . . . . . . . . . . . . . . . . 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 5690	. . . . . . . . . . . . . . . . . . . 20 |- ( H Fn om -> ( H " om ) = ran H )
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( H " om ) = ran H
27	24, 26	syl6eleqr 2559	. . . . . . . . . . . . . . . . . 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 2549	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( aleph ` ( H ` v ) ) e. ( H " om ) )
30		elssuni 4268	. . . . . . . . . . . . . . . 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 3496	. . . . . . . . . . . . . 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 2942	. . . . . . . . . . . 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 2533	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. U. ( H " om ) <-> z e. ( aleph ` z ) ) )
37		eleq2 2533	. . . . . . . . . . . 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 4881	. . . . . . . . . 10 |- ( z e. On -> Ord z )
43	8	onordi 4975	. . . . . . . . . 10 |- Ord ( aleph ` z )
44		ordtri4 4908	. . . . . . . . . 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 2475	. . . . . . . 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 2468	. . . . 5 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> U. ( H " om ) = z )
52	51	fveq2d 5861	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) )
53		eqeq2 2475	. . . . 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 2944	. 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 1374   e. wcel 1762   E.wrex 2808   C_ wss 3469   U.cuni 4238   Ord word 4870   Oncon0 4871   suc csuc 4873   ran crn 4993   |` cres 4994   "cima 4995   Fun wfun 5573   Fn wfn 5574   ` cfv 5579   omcom 6671   reccrdg 7065   cardccrd 8305   alephcale 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-har 7973  df-card 8309  df-aleph 8310

This theorem is referenced by:  alephfp2  8479