Theorem alephsing 8724

Description: The cofinality of a limit aleph is the same as the cofinality of its argument, so if ( aleph ` A ) < A, then ( aleph ` A ) is singular. Conversely, if ( aleph ` A ) is regular (i.e. weakly inaccessible), then ( aleph ` A ) = A, so A has to be rather large (see alephfp 8557). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
alephsing	|- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )

Proof of Theorem alephsing

Dummy variables f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 8514	. . . . . . 7 |- aleph Fn On
2		fnfun 5683	. . . . . . 7 |- ( aleph Fn On -> Fun aleph )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun aleph
4		simpl 464	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. _V )
5		resfunexg 6146	. . . . . 6 |- ( ( Fun aleph /\ A e. _V ) -> ( aleph |` A ) e. _V )
6	3, 4, 5	sylancr 676	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) e. _V )
7		limelon 5493	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> A e. On )
8		onss 6636	. . . . . . . 8 |- ( A e. On -> A C_ On )
9	7, 8	syl 17	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A C_ On )
10		fnssres 5699	. . . . . . 7 |- ( ( aleph Fn On /\ A C_ On ) -> ( aleph |` A ) Fn A )
11	1, 9, 10	sylancr 676	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) Fn A )
12		fvres 5893	. . . . . . . . . . 11 |- ( y e. A -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
13	12	adantl 473	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
14		alephord2i 8526	. . . . . . . . . . 11 |- ( A e. On -> ( y e. A -> ( aleph ` y ) e. ( aleph ` A ) ) )
15	14	imp 436	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( aleph ` y ) e. ( aleph ` A ) )
16	13, 15	eqeltrd 2549	. . . . . . . . 9 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
17	7, 16	sylan 479	. . . . . . . 8 |- ( ( ( A e. _V /\ Lim A ) /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
18	17	ralrimiva 2809	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A. y e. A ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
19		fnfvrnss 6066	. . . . . . 7 |- ( ( ( aleph |` A ) Fn A /\ A. y e. A ( ( aleph |` A ) ` y ) e. ( aleph ` A ) ) -> ran ( aleph |` A ) C_ ( aleph ` A ) )
20	11, 18, 19	syl2anc 673	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ran ( aleph |` A ) C_ ( aleph ` A ) )
21		df-f 5593	. . . . . 6 |- ( ( aleph |` A ) : A --> ( aleph ` A ) <-> ( ( aleph |` A ) Fn A /\ ran ( aleph |` A ) C_ ( aleph ` A ) ) )
22	11, 20, 21	sylanbrc 677	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) : A --> ( aleph ` A ) )
23		alephsmo 8551	. . . . . 6 |- Smo aleph
24		fndm 5685	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
25	1, 24	ax-mp 5	. . . . . . 7 |- dom aleph = On
26	7, 25	syl6eleqr 2560	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. dom aleph )
27		smores 7089	. . . . . 6 |- ( ( Smo aleph /\ A e. dom aleph ) -> Smo ( aleph |` A ) )
28	23, 26, 27	sylancr 676	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> Smo ( aleph |` A ) )
29		alephlim 8516	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> ( aleph ` A ) = U_ y e. A ( aleph ` y ) )
30	29	eleq2d 2534	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) <-> x e. U_ y e. A ( aleph ` y ) ) )
31		eliun 4274	. . . . . . . 8 |- ( x e. U_ y e. A ( aleph ` y ) <-> E. y e. A x e. ( aleph ` y ) )
32		alephon 8518	. . . . . . . . . 10 |- ( aleph ` y ) e. On
33	32	onelssi 5538	. . . . . . . . 9 |- ( x e. ( aleph ` y ) -> x C_ ( aleph ` y ) )
34	33	reximi 2852	. . . . . . . 8 |- ( E. y e. A x e. ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
35	31, 34	sylbi 200	. . . . . . 7 |- ( x e. U_ y e. A ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
36	30, 35	syl6bi 236	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) -> E. y e. A x C_ ( aleph ` y ) ) )
37	36	ralrimiv 2808	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) )
38		feq1 5720	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( f : A --> ( aleph ` A ) <-> ( aleph |` A ) : A --> ( aleph ` A ) ) )
39		smoeq 7087	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( Smo f <-> Smo ( aleph |` A ) ) )
40		fveq1 5878	. . . . . . . . . . . 12 |- ( f = ( aleph |` A ) -> ( f ` y ) = ( ( aleph |` A ) ` y ) )
41	40, 12	sylan9eq 2525	. . . . . . . . . . 11 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( f ` y ) = ( aleph ` y ) )
42	41	sseq2d 3446	. . . . . . . . . 10 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( x C_ ( f ` y ) <-> x C_ ( aleph ` y ) ) )
43	42	rexbidva 2889	. . . . . . . . 9 |- ( f = ( aleph |` A ) -> ( E. y e. A x C_ ( f ` y ) <-> E. y e. A x C_ ( aleph ` y ) ) )
44	43	ralbidv 2829	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) <-> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) ) )
45	38, 39, 44	3anbi123d 1365	. . . . . . 7 |- ( f = ( aleph |` A ) -> ( ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) <-> ( ( aleph |` A ) : A --> ( aleph ` A ) /\ Smo ( aleph |` A ) /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) ) ) )
46	45	spcegv 3121	. . . . . 6 |- ( ( aleph |` A ) e. _V -> ( ( ( aleph |` A ) : A --> ( aleph ` A ) /\ Smo ( aleph |` A ) /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) ) -> E. f ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) ) )
47	46	imp 436	. . . . 5 |- ( ( ( aleph |` A ) e. _V /\ ( ( aleph |` A ) : A --> ( aleph ` A ) /\ Smo ( aleph |` A ) /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) ) ) -> E. f ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) )
48	6, 22, 28, 37, 47	syl13anc 1294	. . . 4 |- ( ( A e. _V /\ Lim A ) -> E. f ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) )
49		alephon 8518	. . . . 5 |- ( aleph ` A ) e. On
50		cfcof 8722	. . . . 5 |- ( ( ( aleph ` A ) e. On /\ A e. On ) -> ( E. f ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
51	49, 7, 50	sylancr 676	. . . 4 |- ( ( A e. _V /\ Lim A ) -> ( E. f ( f : A --> ( aleph ` A ) /\ Smo f /\ A. x e. ( aleph ` A ) E. y e. A x C_ ( f ` y ) ) -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
52	48, 51	mpd 15	. . 3 |- ( ( A e. _V /\ Lim A ) -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
53	52	expcom 442	. 2 |- ( Lim A -> ( A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
54		cf0 8699	. . 3 |- ( cf ` (/) ) = (/)
55		fvprc 5873	. . . 4 |- ( -. A e. _V -> ( aleph ` A ) = (/) )
56	55	fveq2d 5883	. . 3 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` (/) ) )
57		fvprc 5873	. . 3 |- ( -. A e. _V -> ( cf ` A ) = (/) )
58	54, 56, 57	3eqtr4a 2531	. 2 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
59	53, 58	pm2.61d1 164	1 |- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031   C_ wss 3390   (/)c0 3722   U_ciun 4269   dom cdm 4839   ran crn 4840   |` cres 4841   Oncon0 5430   Lim wlim 5431   Fun wfun 5583   Fn wfn 5584   -->wf 5585   ` cfv 5589   Smo wsmo 7082   alephcale 8388   cfccf 8389

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-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-smo 7083  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392  df-cf 8393  df-acn 8394

This theorem is referenced by:  alephom  9028  winafp  9140