Theorem alephfp 8557

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 8558 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 8556	. 2 |- U. ( H " om ) e. ran aleph
3		isinfcard 8541	. . 3 |- ( ( om C_ U. ( H " om ) /\ ( card ` U. ( H " om ) ) = U. ( H " om ) ) <-> U. ( H " om ) e. ran aleph )
4		cardalephex 8539	. . . 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 492	. . 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 218	. 2 |- ( U. ( H " om ) e. ran aleph -> E. z e. On U. ( H " om ) = ( aleph ` z ) )
7		alephle 8537	. . . . . . . . 9 |- ( z e. On -> z C_ ( aleph ` z ) )
8		alephon 8518	. . . . . . . . . . 11 |- ( aleph ` z ) e. On
9	8	onirri 5536	. . . . . . . . . 10 |- -. ( aleph ` z ) e. ( aleph ` z )
10		frfnom 7170	. . . . . . . . . . . . . 14 |- ( rec ( aleph , om ) |` om ) Fn om
11	1	fneq1i 5680	. . . . . . . . . . . . . 14 |- ( H Fn om <-> ( rec ( aleph , om ) |` om ) Fn om )
12	10, 11	mpbir 214	. . . . . . . . . . . . 13 |- H Fn om
13		fnfun 5683	. . . . . . . . . . . . 13 |- ( H Fn om -> Fun H )
14		eluniima 6173	. . . . . . . . . . . . 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 8549	. . . . . . . . . . . . . . . 16 |- ran aleph C_ On
17	1	alephfplem3 8555	. . . . . . . . . . . . . . . 16 |- ( v e. om -> ( H ` v ) e. ran aleph )
18	16, 17	sseldi 3416	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( H ` v ) e. On )
19		alephord2i 8526	. . . . . . . . . . . . . . 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 8554	. . . . . . . . . . . . . . . . 17 |- ( v e. om -> ( H ` suc v ) = ( aleph ` ( H ` v ) ) )
22		peano2 6732	. . . . . . . . . . . . . . . . . 18 |- ( v e. om -> suc v e. om )
23		fnfvelrn 6034	. . . . . . . . . . . . . . . . . . . 20 |- ( ( H Fn om /\ suc v e. om ) -> ( H ` suc v ) e. ran H )
24	12, 23	mpan 684	. . . . . . . . . . . . . . . . . . 19 |- ( suc v e. om -> ( H ` suc v ) e. ran H )
25		fnima 5704	. . . . . . . . . . . . . . . . . . . 20 |- ( H Fn om -> ( H " om ) = ran H )
26	12, 25	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- ( H " om ) = ran H
27	24, 26	syl6eleqr 2560	. . . . . . . . . . . . . . . . . 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 2550	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 15 |- ( v e. om -> ( aleph ` ( H ` v ) ) C_ U. ( H " om ) )
32	31	sseld 3417	. . . . . . . . . . . . . 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 2867	. . . . . . . . . . . 12 |- ( E. v e. om z e. ( H ` v ) -> ( aleph ` z ) e. U. ( H " om ) )
35	15, 34	sylbi 200	. . . . . . . . . . 11 |- ( z e. U. ( H " om ) -> ( aleph ` z ) e. U. ( H " om ) )
36		eleq2 2538	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. U. ( H " om ) <-> z e. ( aleph ` z ) ) )
37		eleq2 2538	. . . . . . . . . . . 12 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` z ) e. U. ( H " om ) <-> ( aleph ` z ) e. ( aleph ` z ) ) )
38	36, 37	imbi12d 327	. . . . . . . . . . 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 216	. . . . . . . . . 10 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z e. ( aleph ` z ) -> ( aleph ` z ) e. ( aleph ` z ) ) )
40	9, 39	mtoi 183	. . . . . . . . 9 |- ( U. ( H " om ) = ( aleph ` z ) -> -. z e. ( aleph ` z ) )
41	7, 40	anim12i 576	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) )
42		eloni 5440	. . . . . . . . . 10 |- ( z e. On -> Ord z )
43	8	onordi 5534	. . . . . . . . . 10 |- Ord ( aleph ` z )
44		ordtri4 5467	. . . . . . . . . 10 |- ( ( Ord z /\ Ord ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
45	42, 43, 44	sylancl 675	. . . . . . . . 9 |- ( z e. On -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
46	45	adantr 472	. . . . . . . 8 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z = ( aleph ` z ) <-> ( z C_ ( aleph ` z ) /\ -. z e. ( aleph ` z ) ) ) )
47	41, 46	mpbird 240	. . . . . . 7 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> z = ( aleph ` z ) )
48		eqeq2 2482	. . . . . . . 8 |- ( U. ( H " om ) = ( aleph ` z ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
49	48	adantl 473	. . . . . . 7 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( z = U. ( H " om ) <-> z = ( aleph ` z ) ) )
50	47, 49	mpbird 240	. . . . . 6 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> z = U. ( H " om ) )
51	50	eqcomd 2477	. . . . 5 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> U. ( H " om ) = z )
52	51	fveq2d 5883	. . . 4 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) )
53		eqeq2 2482	. . . . 5 |- ( U. ( H " om ) = ( aleph ` z ) -> ( ( aleph ` U. ( H " om ) ) = U. ( H " om ) <-> ( aleph ` U. ( H " om ) ) = ( aleph ` z ) ) )
54	53	adantl 473	. . . 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 240	. . 3 |- ( ( z e. On /\ U. ( H " om ) = ( aleph ` z ) ) -> ( aleph ` U. ( H " om ) ) = U. ( H " om ) )
56	55	rexlimiva 2868	. 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 189   /\ wa 376   = wceq 1452   e. wcel 1904   E.wrex 2757   C_ wss 3390   U.cuni 4190   ran crn 4840   |` cres 4841   "cima 4842   Ord word 5429   Oncon0 5430   suc csuc 5432   Fun wfun 5583   Fn wfn 5584   ` cfv 5589   omcom 6711   reccrdg 7145   cardccrd 8387   alephcale 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392

This theorem is referenced by:  alephfp2  8558