Theorem alephsing 9098

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 8931). 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 8888	. . . . . . 7 |- aleph Fn On
2		fnfun 5988	. . . . . . 7 |- ( aleph Fn On -> Fun aleph )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun aleph
4		simpl 473	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. _V )
5		resfunexg 6479	. . . . . 6 |- ( ( Fun aleph /\ A e. _V ) -> ( aleph |` A ) e. _V )
6	3, 4, 5	sylancr 695	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) e. _V )
7		limelon 5788	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> A e. On )
8		onss 6990	. . . . . . . 8 |- ( A e. On -> A C_ On )
9	7, 8	syl 17	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A C_ On )
10		fnssres 6004	. . . . . . 7 |- ( ( aleph Fn On /\ A C_ On ) -> ( aleph |` A ) Fn A )
11	1, 9, 10	sylancr 695	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) Fn A )
12		fvres 6207	. . . . . . . . . . 11 |- ( y e. A -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
13	12	adantl 482	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
14		alephord2i 8900	. . . . . . . . . . 11 |- ( A e. On -> ( y e. A -> ( aleph ` y ) e. ( aleph ` A ) ) )
15	14	imp 445	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( aleph ` y ) e. ( aleph ` A ) )
16	13, 15	eqeltrd 2701	. . . . . . . . 9 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
17	7, 16	sylan 488	. . . . . . . 8 |- ( ( ( A e. _V /\ Lim A ) /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
18	17	ralrimiva 2966	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A. y e. A ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
19		fnfvrnss 6390	. . . . . . 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 693	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ran ( aleph |` A ) C_ ( aleph ` A ) )
21		df-f 5892	. . . . . 6 |- ( ( aleph |` A ) : A --> ( aleph ` A ) <-> ( ( aleph |` A ) Fn A /\ ran ( aleph |` A ) C_ ( aleph ` A ) ) )
22	11, 20, 21	sylanbrc 698	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) : A --> ( aleph ` A ) )
23		alephsmo 8925	. . . . . 6 |- Smo aleph
24		fndm 5990	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
25	1, 24	ax-mp 5	. . . . . . 7 |- dom aleph = On
26	7, 25	syl6eleqr 2712	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. dom aleph )
27		smores 7449	. . . . . 6 |- ( ( Smo aleph /\ A e. dom aleph ) -> Smo ( aleph |` A ) )
28	23, 26, 27	sylancr 695	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> Smo ( aleph |` A ) )
29		alephlim 8890	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> ( aleph ` A ) = U_ y e. A ( aleph ` y ) )
30	29	eleq2d 2687	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) <-> x e. U_ y e. A ( aleph ` y ) ) )
31		eliun 4524	. . . . . . . 8 |- ( x e. U_ y e. A ( aleph ` y ) <-> E. y e. A x e. ( aleph ` y ) )
32		alephon 8892	. . . . . . . . . 10 |- ( aleph ` y ) e. On
33	32	onelssi 5836	. . . . . . . . 9 |- ( x e. ( aleph ` y ) -> x C_ ( aleph ` y ) )
34	33	reximi 3011	. . . . . . . 8 |- ( E. y e. A x e. ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
35	31, 34	sylbi 207	. . . . . . 7 |- ( x e. U_ y e. A ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
36	30, 35	syl6bi 243	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) -> E. y e. A x C_ ( aleph ` y ) ) )
37	36	ralrimiv 2965	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) )
38		feq1 6026	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( f : A --> ( aleph ` A ) <-> ( aleph |` A ) : A --> ( aleph ` A ) ) )
39		smoeq 7447	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( Smo f <-> Smo ( aleph |` A ) ) )
40		fveq1 6190	. . . . . . . . . . . 12 |- ( f = ( aleph |` A ) -> ( f ` y ) = ( ( aleph |` A ) ` y ) )
41	40, 12	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( f ` y ) = ( aleph ` y ) )
42	41	sseq2d 3633	. . . . . . . . . 10 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( x C_ ( f ` y ) <-> x C_ ( aleph ` y ) ) )
43	42	rexbidva 3049	. . . . . . . . 9 |- ( f = ( aleph |` A ) -> ( E. y e. A x C_ ( f ` y ) <-> E. y e. A x C_ ( aleph ` y ) ) )
44	43	ralbidv 2986	. . . . . . . 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 1399	. . . . . . 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 3294	. . . . . 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 445	. . . . 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 1328	. . . 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 8892	. . . . 5 |- ( aleph ` A ) e. On
50		cfcof 9096	. . . . 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 695	. . . 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 451	. 2 |- ( Lim A -> ( A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
54		cf0 9073	. . 3 |- ( cf ` (/) ) = (/)
55		fvprc 6185	. . . 4 |- ( -. A e. _V -> ( aleph ` A ) = (/) )
56	55	fveq2d 6195	. . 3 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` (/) ) )
57		fvprc 6185	. . 3 |- ( -. A e. _V -> ( cf ` A ) = (/) )
58	54, 56, 57	3eqtr4a 2682	. 2 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
59	53, 58	pm2.61d1 171	1 |- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U_ciun 4520   dom cdm 5114   ran crn 5115   |` cres 5116   Oncon0 5723   Lim wlim 5724   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   Smo wsmo 7442   alephcale 8762   cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768

This theorem is referenced by:  alephom  9407  winafp  9519