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Theorem mappwen 7739
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 733 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ~P B
)
2 pw2eng 6968 . . . . . 6  |-  ( B  e.  dom  card  ->  ~P B  ~~  ( 2o 
^m  B ) )
32ad2antrr 706 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~~  ( 2o  ^m  B ) )
4 domentr 6920 . . . . 5  |-  ( ( A  ~<_  ~P B  /\  ~P B  ~~  ( 2o  ^m  B ) )  ->  A  ~<_  ( 2o  ^m  B ) )
51, 3, 4syl2anc 642 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ( 2o 
^m  B ) )
6 mapdom1 7026 . . . 4  |-  ( A  ~<_  ( 2o  ^m  B
)  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
75, 6syl 15 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
8 2on 6487 . . . . . . 7  |-  2o  e.  On
98a1i 10 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  2o  e.  On )
10 simpll 730 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  B  e.  dom  card )
11 mapxpen 7027 . . . . . 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 1182 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
138elexi 2797 . . . . . . 7  |-  2o  e.  _V
1413enref 6894 . . . . . 6  |-  2o  ~~  2o
15 infxpidm2 7644 . . . . . . 7  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B )  -> 
( B  X.  B
)  ~~  B )
1615adantr 451 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( B  X.  B )  ~~  B
)
17 mapen 7025 . . . . . 6  |-  ( ( 2o  ~~  2o  /\  ( B  X.  B
)  ~~  B )  ->  ( 2o  ^m  ( B  X.  B ) ) 
~~  ( 2o  ^m  B ) )
1814, 16, 17sylancr 644 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  ( B  X.  B
) )  ~~  ( 2o  ^m  B ) )
19 entr 6913 . . . . 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 642 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
21 ensym 6910 . . . . 5  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
223, 21syl 15 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~~  ~P B )
23 entr 6913 . . . 4  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )
2420, 22, 23syl2anc 642 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ~P B )
25 domentr 6920 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  ( ( 2o 
^m  B )  ^m  B )  /\  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )  -> 
( A  ^m  B
)  ~<_  ~P B )
267, 24, 25syl2anc 642 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ~P B
)
27 mapdom1 7026 . . . 4  |-  ( 2o  ~<_  A  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
2827ad2antrl 708 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
29 endomtr 6919 . . 3  |-  ( ( ~P B  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~<_  ( A  ^m  B
) )  ->  ~P B  ~<_  ( A  ^m  B ) )
303, 28, 29syl2anc 642 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~<_  ( A  ^m  B ) )
31 sbth 6981 . 2  |-  ( ( ( A  ^m  B
)  ~<_  ~P B  /\  ~P B  ~<_  ( A  ^m  B ) )  -> 
( A  ^m  B
)  ~~  ~P B
)
3226, 30, 31syl2anc 642 1  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861   cardccrd 7568
This theorem is referenced by:  alephexp1  8201  hauspwdom  17227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572
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