Theorem alephsing 8656

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 8489). 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 8446	. . . . . . 7 |- aleph Fn On
2		fnfun 5678	. . . . . . 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 6126	. . . . . 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 4941	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> A e. On )
8		onss 6610	. . . . . . . 8 |- ( A e. On -> A C_ On )
9	7, 8	syl 16	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A C_ On )
10		fnssres 5694	. . . . . . 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 5880	. . . . . . . . . . 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 8458	. . . . . . . . . . 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 2555	. . . . . . . . 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 2878	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A. y e. A ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
19		fnfvrnss 6049	. . . . . . 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 5592	. . . . . 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 8483	. . . . . 6 |- Smo aleph
24		fndm 5680	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
25	1, 24	ax-mp 5	. . . . . . 7 |- dom aleph = On
26	7, 25	syl6eleqr 2566	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. dom aleph )
27		smores 7023	. . . . . 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 8448	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> ( aleph ` A ) = U_ y e. A ( aleph ` y ) )
30	29	eleq2d 2537	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) <-> x e. U_ y e. A ( aleph ` y ) ) )
31		eliun 4330	. . . . . . . 8 |- ( x e. U_ y e. A ( aleph ` y ) <-> E. y e. A x e. ( aleph ` y ) )
32		alephon 8450	. . . . . . . . . 10 |- ( aleph ` y ) e. On
33	32	onelssi 4986	. . . . . . . . 9 |- ( x e. ( aleph ` y ) -> x C_ ( aleph ` y ) )
34	33	reximi 2932	. . . . . . . 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 2876	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) )
38		feq1 5713	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( f : A --> ( aleph ` A ) <-> ( aleph |` A ) : A --> ( aleph ` A ) ) )
39		smoeq 7021	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( Smo f <-> Smo ( aleph |` A ) ) )
40		fveq1 5865	. . . . . . . . . . . 12 |- ( f = ( aleph |` A ) -> ( f ` y ) = ( ( aleph |` A ) ` y ) )
41	40, 12	sylan9eq 2528	. . . . . . . . . . 11 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( f ` y ) = ( aleph ` y ) )
42	41	sseq2d 3532	. . . . . . . . . 10 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( x C_ ( f ` y ) <-> x C_ ( aleph ` y ) ) )
43	42	rexbidva 2970	. . . . . . . . 9 |- ( f = ( aleph |` A ) -> ( E. y e. A x C_ ( f ` y ) <-> E. y e. A x C_ ( aleph ` y ) ) )
44	43	ralbidv 2903	. . . . . . . 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 1299	. . . . . . 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 3199	. . . . . 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 1230	. . . 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 8450	. . . . 5 |- ( aleph ` A ) e. On
50		cfcof 8654	. . . . 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 8631	. . 3 |- ( cf ` (/) ) = (/)
55		fvprc 5860	. . . 4 |- ( -. A e. _V -> ( aleph ` A ) = (/) )
56	55	fveq2d 5870	. . 3 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` (/) ) )
57		fvprc 5860	. . 3 |- ( -. A e. _V -> ( cf ` A ) = (/) )
58	54, 56, 57	3eqtr4a 2534	. 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 973   = wceq 1379   E.wex 1596   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   C_ wss 3476   (/)c0 3785   U_ciun 4325   Oncon0 4878   Lim wlim 4879   dom cdm 4999   ran crn 5000   |` cres 5001   Fun wfun 5582   Fn wfn 5583   -->wf 5584   ` cfv 5588   Smo wsmo 7016   alephcale 8317   cfccf 8318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-smo 7017  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-har 7984  df-card 8320  df-aleph 8321  df-cf 8322  df-acn 8323

This theorem is referenced by:  alephom  8960  winafp  9075