Theorem alephsing 8704

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 8537). 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 8494	. . . . . . 7 |- aleph Fn On
2		fnfun 5691	. . . . . . 7 |- ( aleph Fn On -> Fun aleph )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun aleph
4		simpl 458	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. _V )
5		resfunexg 6145	. . . . . 6 |- ( ( Fun aleph /\ A e. _V ) -> ( aleph |` A ) e. _V )
6	3, 4, 5	sylancr 667	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) e. _V )
7		limelon 5505	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> A e. On )
8		onss 6631	. . . . . . . 8 |- ( A e. On -> A C_ On )
9	7, 8	syl 17	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> A C_ On )
10		fnssres 5707	. . . . . . 7 |- ( ( aleph Fn On /\ A C_ On ) -> ( aleph |` A ) Fn A )
11	1, 9, 10	sylancr 667	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) Fn A )
12		fvres 5895	. . . . . . . . . . 11 |- ( y e. A -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
13	12	adantl 467	. . . . . . . . . 10 |- ( ( A e. On /\ y e. A ) -> ( ( aleph |` A ) ` y ) = ( aleph ` y ) )
14		alephord2i 8506	. . . . . . . . . . 11 |- ( A e. On -> ( y e. A -> ( aleph ` y ) e. ( aleph ` A ) ) )
15	14	imp 430	. . . . . . . . . 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 473	. . . . . . . 8 |- ( ( ( A e. _V /\ Lim A ) /\ y e. A ) -> ( ( aleph |` A ) ` y ) e. ( aleph ` A ) )
18	17	ralrimiva 2846	. . . . . . 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 665	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ran ( aleph |` A ) C_ ( aleph ` A ) )
21		df-f 5605	. . . . . 6 |- ( ( aleph |` A ) : A --> ( aleph ` A ) <-> ( ( aleph |` A ) Fn A /\ ran ( aleph |` A ) C_ ( aleph ` A ) ) )
22	11, 20, 21	sylanbrc 668	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> ( aleph |` A ) : A --> ( aleph ` A ) )
23		alephsmo 8531	. . . . . 6 |- Smo aleph
24		fndm 5693	. . . . . . . 8 |- ( aleph Fn On -> dom aleph = On )
25	1, 24	ax-mp 5	. . . . . . 7 |- dom aleph = On
26	7, 25	syl6eleqr 2528	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> A e. dom aleph )
27		smores 7079	. . . . . 6 |- ( ( Smo aleph /\ A e. dom aleph ) -> Smo ( aleph |` A ) )
28	23, 26, 27	sylancr 667	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> Smo ( aleph |` A ) )
29		alephlim 8496	. . . . . . . 8 |- ( ( A e. _V /\ Lim A ) -> ( aleph ` A ) = U_ y e. A ( aleph ` y ) )
30	29	eleq2d 2499	. . . . . . 7 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) <-> x e. U_ y e. A ( aleph ` y ) ) )
31		eliun 4307	. . . . . . . 8 |- ( x e. U_ y e. A ( aleph ` y ) <-> E. y e. A x e. ( aleph ` y ) )
32		alephon 8498	. . . . . . . . . 10 |- ( aleph ` y ) e. On
33	32	onelssi 5550	. . . . . . . . 9 |- ( x e. ( aleph ` y ) -> x C_ ( aleph ` y ) )
34	33	reximi 2900	. . . . . . . 8 |- ( E. y e. A x e. ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
35	31, 34	sylbi 198	. . . . . . 7 |- ( x e. U_ y e. A ( aleph ` y ) -> E. y e. A x C_ ( aleph ` y ) )
36	30, 35	syl6bi 231	. . . . . 6 |- ( ( A e. _V /\ Lim A ) -> ( x e. ( aleph ` A ) -> E. y e. A x C_ ( aleph ` y ) ) )
37	36	ralrimiv 2844	. . . . 5 |- ( ( A e. _V /\ Lim A ) -> A. x e. ( aleph ` A ) E. y e. A x C_ ( aleph ` y ) )
38		feq1 5728	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( f : A --> ( aleph ` A ) <-> ( aleph |` A ) : A --> ( aleph ` A ) ) )
39		smoeq 7077	. . . . . . . 8 |- ( f = ( aleph |` A ) -> ( Smo f <-> Smo ( aleph |` A ) ) )
40		fveq1 5880	. . . . . . . . . . . 12 |- ( f = ( aleph |` A ) -> ( f ` y ) = ( ( aleph |` A ) ` y ) )
41	40, 12	sylan9eq 2490	. . . . . . . . . . 11 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( f ` y ) = ( aleph ` y ) )
42	41	sseq2d 3498	. . . . . . . . . 10 |- ( ( f = ( aleph |` A ) /\ y e. A ) -> ( x C_ ( f ` y ) <-> x C_ ( aleph ` y ) ) )
43	42	rexbidva 2943	. . . . . . . . 9 |- ( f = ( aleph |` A ) -> ( E. y e. A x C_ ( f ` y ) <-> E. y e. A x C_ ( aleph ` y ) ) )
44	43	ralbidv 2871	. . . . . . . 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 1335	. . . . . . 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 3173	. . . . . 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 430	. . . . 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 1266	. . . 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 8498	. . . . 5 |- ( aleph ` A ) e. On
50		cfcof 8702	. . . . 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 667	. . . 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 436	. 2 |- ( Lim A -> ( A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) ) )
54		cf0 8679	. . 3 |- ( cf ` (/) ) = (/)
55		fvprc 5875	. . . 4 |- ( -. A e. _V -> ( aleph ` A ) = (/) )
56	55	fveq2d 5885	. . 3 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` (/) ) )
57		fvprc 5875	. . 3 |- ( -. A e. _V -> ( cf ` A ) = (/) )
58	54, 56, 57	3eqtr4a 2496	. 2 |- ( -. A e. _V -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )
59	53, 58	pm2.61d1 162	1 |- ( Lim A -> ( cf ` ( aleph ` A ) ) = ( cf ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1659   e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087   C_ wss 3442   (/)c0 3767   U_ciun 4302   dom cdm 4854   ran crn 4855   |` cres 4856   Oncon0 5442   Lim wlim 5443   Fun wfun 5595   Fn wfn 5596   -->wf 5597   ` cfv 5601   Smo wsmo 7072   alephcale 8369   cfccf 8370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-smo 7073  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-har 8073  df-card 8372  df-aleph 8373  df-cf 8374  df-acn 8375

This theorem is referenced by:  alephom  9008  winafp  9121