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Theorem mappwen 8282
Description: Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
mappwen  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )

Proof of Theorem mappwen
StepHypRef Expression
1 simprr 756 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ~P B
)
2 pw2eng 7417 . . . . . 6  |-  ( B  e.  dom  card  ->  ~P B  ~~  ( 2o 
^m  B ) )
32ad2antrr 725 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~~  ( 2o  ^m  B ) )
4 domentr 7368 . . . . 5  |-  ( ( A  ~<_  ~P B  /\  ~P B  ~~  ( 2o  ^m  B ) )  ->  A  ~<_  ( 2o  ^m  B ) )
51, 3, 4syl2anc 661 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ( 2o 
^m  B ) )
6 mapdom1 7476 . . . 4  |-  ( A  ~<_  ( 2o  ^m  B
)  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
75, 6syl 16 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
8 2on 6928 . . . . . . 7  |-  2o  e.  On
98a1i 11 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  2o  e.  On )
10 simpll 753 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  B  e.  dom  card )
11 mapxpen 7477 . . . . . 6  |-  ( ( 2o  e.  On  /\  B  e.  dom  card  /\  B  e.  dom  card )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
129, 10, 10, 11syl3anc 1218 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
138elexi 2982 . . . . . . 7  |-  2o  e.  _V
1413enref 7342 . . . . . 6  |-  2o  ~~  2o
15 infxpidm2 8183 . . . . . . 7  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B )  -> 
( B  X.  B
)  ~~  B )
1615adantr 465 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( B  X.  B )  ~~  B
)
17 mapen 7475 . . . . . 6  |-  ( ( 2o  ~~  2o  /\  ( B  X.  B
)  ~~  B )  ->  ( 2o  ^m  ( B  X.  B ) ) 
~~  ( 2o  ^m  B ) )
1814, 16, 17sylancr 663 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  ( B  X.  B
) )  ~~  ( 2o  ^m  B ) )
19 entr 7361 . . . . 5  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  ( B  X.  B
) )  /\  ( 2o  ^m  ( B  X.  B ) )  ~~  ( 2o  ^m  B ) )  ->  ( ( 2o  ^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
2012, 18, 19syl2anc 661 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
213ensymd 7360 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~~  ~P B )
22 entr 7361 . . . 4  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )
2320, 21, 22syl2anc 661 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ~P B )
24 domentr 7368 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  ( ( 2o 
^m  B )  ^m  B )  /\  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )  -> 
( A  ^m  B
)  ~<_  ~P B )
257, 23, 24syl2anc 661 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 mapdom1 7476 . . . 4  |-  ( 2o  ~<_  A  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
2726ad2antrl 727 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
28 endomtr 7367 . . 3  |-  ( ( ~P B  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~<_  ( A  ^m  B
) )  ->  ~P B  ~<_  ( A  ^m  B ) )
293, 27, 28syl2anc 661 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~<_  ( A  ^m  B ) )
30 sbth 7431 . 2  |-  ( ( ( A  ^m  B
)  ~<_  ~P B  /\  ~P B  ~<_  ( A  ^m  B ) )  -> 
( A  ^m  B
)  ~~  ~P B
)
3125, 29, 30syl2anc 661 1  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   ~Pcpw 3860   class class class wbr 4292   Oncon0 4719    X. cxp 4838   dom cdm 4840  (class class class)co 6091   omcom 6476   2oc2o 6914    ^m cmap 7214    ~~ cen 7307    ~<_ cdom 7308   cardccrd 8105
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-oi 7724  df-card 8109
This theorem is referenced by:  alephexp1  8743  hauspwdom  19105
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