Theorem hauspwdom 21304

Description: Simplify the cardinal A ^ NN of hausmapdom 21303 to ~P B = 2 ^ B when B is an infinite cardinal greater than A. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypothesis

Ref	Expression
hauspwdom.1	|- X = U. J

Assertion

Ref	Expression
hauspwdom	|- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( ( cls ` J ) ` A ) ~<_ ~P B )

Proof of Theorem hauspwdom

Step	Hyp	Ref	Expression
1		hauspwdom.1	. . . 4 |- X = U. J
2	1	hausmapdom 21303	. . 3 |- ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) -> ( ( cls ` J ) ` A ) ~<_ ( A ^m NN ) )
3	2	adantr 481	. 2 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( ( cls ` J ) ` A ) ~<_ ( A ^m NN ) )
4		simprr 796	. . . 4 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> NN ~<_ B )
5		1nn 11031	. . . . 5 |- 1 e. NN
6		noel 3919	. . . . . . 7 |- -. 1 e. (/)
7		eleq2 2690	. . . . . . 7 |- ( NN = (/) -> ( 1 e. NN <-> 1 e. (/) ) )
8	6, 7	mtbiri 317	. . . . . 6 |- ( NN = (/) -> -. 1 e. NN )
9	8	adantr 481	. . . . 5 |- ( ( NN = (/) /\ A = (/) ) -> -. 1 e. NN )
10	5, 9	mt2 191	. . . 4 |- -. ( NN = (/) /\ A = (/) )
11		mapdom2 8131	. . . 4 |- ( ( NN ~<_ B /\ -. ( NN = (/) /\ A = (/) ) ) -> ( A ^m NN ) ~<_ ( A ^m B ) )
12	4, 10, 11	sylancl 694	. . 3 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( A ^m NN ) ~<_ ( A ^m B ) )
13		sdomdom 7983	. . . . . . 7 |- ( A ~< 2o -> A ~<_ 2o )
14	13	adantl 482	. . . . . 6 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ A ~< 2o ) -> A ~<_ 2o )
15		mapdom1 8125	. . . . . 6 |- ( A ~<_ 2o -> ( A ^m B ) ~<_ ( 2o ^m B ) )
16	14, 15	syl 17	. . . . 5 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ A ~< 2o ) -> ( A ^m B ) ~<_ ( 2o ^m B ) )
17		reldom 7961	. . . . . . . . 9 |- Rel ~<_
18	17	brrelex2i 5159	. . . . . . . 8 |- ( NN ~<_ B -> B e. _V )
19	18	ad2antll 765	. . . . . . 7 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> B e. _V )
20		pw2eng 8066	. . . . . . 7 |- ( B e. _V -> ~P B ~~ ( 2o ^m B ) )
21		ensym 8005	. . . . . . 7 |- ( ~P B ~~ ( 2o ^m B ) -> ( 2o ^m B ) ~~ ~P B )
22	19, 20, 21	3syl 18	. . . . . 6 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( 2o ^m B ) ~~ ~P B )
23	22	adantr 481	. . . . 5 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ A ~< 2o ) -> ( 2o ^m B ) ~~ ~P B )
24		domentr 8015	. . . . 5 |- ( ( ( A ^m B ) ~<_ ( 2o ^m B ) /\ ( 2o ^m B ) ~~ ~P B ) -> ( A ^m B ) ~<_ ~P B )
25	16, 23, 24	syl2anc 693	. . . 4 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ A ~< 2o ) -> ( A ^m B ) ~<_ ~P B )
26		onfin2 8152	. . . . . . . . 9 |- om = ( On i^i Fin )
27		inss2 3834	. . . . . . . . 9 |- ( On i^i Fin ) C_ Fin
28	26, 27	eqsstri 3635	. . . . . . . 8 |- om C_ Fin
29		2onn 7720	. . . . . . . 8 |- 2o e. om
30	28, 29	sselii 3600	. . . . . . 7 |- 2o e. Fin
31		simprl 794	. . . . . . . 8 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> A ~<_ ~P B )
32	17	brrelexi 5158	. . . . . . . 8 |- ( A ~<_ ~P B -> A e. _V )
33	31, 32	syl 17	. . . . . . 7 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> A e. _V )
34		fidomtri 8819	. . . . . . 7 |- ( ( 2o e. Fin /\ A e. _V ) -> ( 2o ~<_ A <-> -. A ~< 2o ) )
35	30, 33, 34	sylancr 695	. . . . . 6 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( 2o ~<_ A <-> -. A ~< 2o ) )
36	35	biimpar 502	. . . . 5 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ -. A ~< 2o ) -> 2o ~<_ A )
37		numth3 9292	. . . . . . . . 9 |- ( B e. _V -> B e. dom card )
38	19, 37	syl 17	. . . . . . . 8 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> B e. dom card )
39	38	adantr 481	. . . . . . 7 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> B e. dom card )
40		nnenom 12779	. . . . . . . . . 10 |- NN ~~ om
41	40	ensymi 8006	. . . . . . . . 9 |- om ~~ NN
42		endomtr 8014	. . . . . . . . 9 |- ( ( om ~~ NN /\ NN ~<_ B ) -> om ~<_ B )
43	41, 4, 42	sylancr 695	. . . . . . . 8 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> om ~<_ B )
44	43	adantr 481	. . . . . . 7 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> om ~<_ B )
45		simpr 477	. . . . . . 7 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> 2o ~<_ A )
46	31	adantr 481	. . . . . . 7 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> A ~<_ ~P B )
47		mappwen 8935	. . . . . . 7 |- ( ( ( B e. dom card /\ om ~<_ B ) /\ ( 2o ~<_ A /\ A ~<_ ~P B ) ) -> ( A ^m B ) ~~ ~P B )
48	39, 44, 45, 46, 47	syl22anc 1327	. . . . . 6 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> ( A ^m B ) ~~ ~P B )
49		endom 7982	. . . . . 6 |- ( ( A ^m B ) ~~ ~P B -> ( A ^m B ) ~<_ ~P B )
50	48, 49	syl 17	. . . . 5 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ 2o ~<_ A ) -> ( A ^m B ) ~<_ ~P B )
51	36, 50	syldan 487	. . . 4 |- ( ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) /\ -. A ~< 2o ) -> ( A ^m B ) ~<_ ~P B )
52	25, 51	pm2.61dan 832	. . 3 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( A ^m B ) ~<_ ~P B )
53		domtr 8009	. . 3 |- ( ( ( A ^m NN ) ~<_ ( A ^m B ) /\ ( A ^m B ) ~<_ ~P B ) -> ( A ^m NN ) ~<_ ~P B )
54	12, 52, 53	syl2anc 693	. 2 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( A ^m NN ) ~<_ ~P B )
55		domtr 8009	. 2 |- ( ( ( ( cls ` J ) ` A ) ~<_ ( A ^m NN ) /\ ( A ^m NN ) ~<_ ~P B ) -> ( ( cls ` J ) ` A ) ~<_ ~P B )
56	3, 54, 55	syl2anc 693	1 |- ( ( ( J e. Haus /\ J e. 1stc /\ A C_ X ) /\ ( A ~<_ ~P B /\ NN ~<_ B ) ) -> ( ( cls ` J ) ` A ) ~<_ ~P B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888  (class class class)co 6650   omcom 7065   2oc2o 7554   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   cardccrd 8761   1c1 9937   NNcn 11020   clsccl 20822   Hauscha 21112   1stcc1stc 21240

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  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  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-iin 4523  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-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033  df-haus 21119  df-1stc 21242

This theorem is referenced by: (None)