Theorem alephsing 8450

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 8283). 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 8240	. . . . . . 7 |- aleph Fn On
2		fnfun 5513	. . . . . . 7 |- ( aleph Fn On -> Fun aleph )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun aleph
4		simpl 457	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. _V )
5		resfunexg 5948	. . . . . 6 |- ( ( Fun aleph /\ A e. _V ) -> ( aleph |` A ) e. _V )
6	3, 4, 5	sylancr 663	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) e. _V )
7		limelon 4787	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> A e. On )
8		onss 6407	. . . . . . . 8 |- ( A e. On -> A C_ On )
9	7, 8	syl 16	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A C_ On )
10		fnssres 5529	. . . . . . 7 |- ( ( aleph Fn On /\ A C_ On ) -> ( aleph |` A ) Fn A )
11	1, 9, 10	sylancr 663	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) Fn A )
12		fvres 5709	. . . . . . . . . . 11 |- ( y e. A -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
13	12	adantl 466	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
14		alephord2i 8252	. . . . . . . . . . 11 |- ( A e. On -> ( y e. A -> ( aleph ` y ) e. ( aleph ` A ) ) )
15	14	imp 429	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( aleph ` y ) e. ( aleph ` A ) )
16	13, 15	eqeltrd 2517	. . . . . . . . 9 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
17	7, 16	sylan 471	. . . . . . . 8 |- ( ( ( A e. _V /\ Lim A ) /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
18	17	ralrimiva 2804	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A. y e. A ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
19		fnfvrnss 5876	. . . . . . 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 661	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ran ( aleph |` A ) C_ ( aleph ` A ) )
21		df-f 5427	. . . . . 6 |- ( ( aleph |` A ) : A --> ( aleph ` A ) <-> ( ( aleph |` A ) Fn A /\ ran ( aleph |` A ) C_ ( aleph ` A ) ) )
22	11, 20, 21	sylanbrc 664	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) : A --> ( aleph ` A ) )
23		alephsmo 8277	. . . . . 6 |- Smo aleph
24		fndm 5515	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
25	1, 24	ax-mp 5	. . . . . . 7 |- dom aleph = On
26	7, 25	syl6eleqr 2534	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. dom aleph )
27		smores 6818	. . . . . 6 |- ( ( Smo aleph /\ A e. dom aleph ) -> Smo ( aleph |` A ) )
28	23, 26, 27	sylancr 663	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> Smo ( aleph |` A ) )
29		alephlim 8242	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> ( aleph ` A ) = U_ y e. A ( aleph ` y ) )
30	29	eleq2d 2510	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) <-> x e. U_ y e. A ( aleph ` y ) ) )
31		eliun 4180	. . . . . . . 8 |- ( x e. U_ y e. A ( aleph ` y ) <-> E. y e. A x e. ( aleph ` y ) )
32		alephon 8244	. . . . . . . . . 10 |- ( aleph ` y ) e. On
33	32	onelssi 4832	. . . . . . . . 9 |- ( x e. ( aleph ` y ) -> x C_ ( aleph ` y ) )
34	33	reximi 2828	. . . . . . . 8 |- ( E. y e. A x e. ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
35	31, 34	sylbi 195	. . . . . . 7 |- ( x e. U_ y e. A ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
36	30, 35	syl6bi 228	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) -> E. y e. A x C_ ( aleph ` y ) ) )
37	36	ralrimiv 2803	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) )
38		feq1 5547	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( f : A --> ( aleph ` A ) <-> ( aleph |` A ) : A --> ( aleph ` A ) ) )
39		smoeq 6816	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( Smo f <-> Smo ( aleph |` A ) ) )
40		fveq1 5695	. . . . . . . . . . . 12 |- ( f = ( aleph |` A ) -> ( f ` y ) = ( ( aleph |` A ) ` y ) )
41	40, 12	sylan9eq 2495	. . . . . . . . . . 11 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( f ` y ) = ( aleph ` y ) )
42	41	sseq2d 3389	. . . . . . . . . 10 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( x C_ ( f ` y ) <-> x C_ ( aleph ` y ) ) )
43	42	rexbidva 2737	. . . . . . . . 9 |- ( f = ( aleph |` A ) -> ( E. y e. A x C_ ( f ` y ) <-> E. y e. A x C_ ( aleph ` y ) ) )
44	43	ralbidv 2740	. . . . . . . 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 1289	. . . . . . 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 3063	. . . . . 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 429	. . . . 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 1220	. . . 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 8244	. . . . 5 |- ( aleph ` A ) e. On
50		cfcof 8448	. . . . 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 663	. . . 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 435	. 2 |- ( Lim A -> ( A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
54		cf0 8425	. . 3 |- ( cf ` (/) ) = (/)
55		fvprc 5690	. . . 4 |- ( -. A e. _V -> ( aleph ` A ) = (/) )
56	55	fveq2d 5700	. . 3 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` (/) ) )
57		fvprc 5690	. . 3 |- ( -. A e. _V -> ( cf ` A ) = (/) )
58	54, 56, 57	3eqtr4a 2501	. 2 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
59	53, 58	pm2.61d1 159	1 |- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977   C_ wss 3333   (/)c0 3642   U_ciun 4176   Oncon0 4724   Lim wlim 4725   dom cdm 4845   ran crn 4846   |` cres 4847   Fun wfun 5417   Fn wfn 5418   -->wf 5419   ` cfv 5423   Smo wsmo 6811   alephcale 8111   cfccf 8112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-smo 6812  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-har 7778  df-card 8114  df-aleph 8115  df-cf 8116  df-acn 8117

This theorem is referenced by:  alephom  8754  winafp  8869