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Theorem mappwen 8525
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 758 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ~P B
)
2 pw2eng 7661 . . . . . 6  |-  ( B  e.  dom  card  ->  ~P B  ~~  ( 2o 
^m  B ) )
32ad2antrr 724 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~~  ( 2o  ^m  B ) )
4 domentr 7612 . . . . 5  |-  ( ( A  ~<_  ~P B  /\  ~P B  ~~  ( 2o  ^m  B ) )  ->  A  ~<_  ( 2o  ^m  B ) )
51, 3, 4syl2anc 659 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ( 2o 
^m  B ) )
6 mapdom1 7720 . . . 4  |-  ( A  ~<_  ( 2o  ^m  B
)  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
75, 6syl 17 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ( ( 2o  ^m  B )  ^m  B ) )
8 2on 7175 . . . . . . 7  |-  2o  e.  On
98a1i 11 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  2o  e.  On )
10 simpll 752 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  B  e.  dom  card )
11 mapxpen 7721 . . . . . 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 1230 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
138elexi 3069 . . . . . . 7  |-  2o  e.  _V
1413enref 7586 . . . . . 6  |-  2o  ~~  2o
15 infxpidm2 8426 . . . . . . 7  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B )  -> 
( B  X.  B
)  ~~  B )
1615adantr 463 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( B  X.  B )  ~~  B
)
17 mapen 7719 . . . . . 6  |-  ( ( 2o  ~~  2o  /\  ( B  X.  B
)  ~~  B )  ->  ( 2o  ^m  ( B  X.  B ) ) 
~~  ( 2o  ^m  B ) )
1814, 16, 17sylancr 661 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  ( B  X.  B
) )  ~~  ( 2o  ^m  B ) )
19 entr 7605 . . . . 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 659 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
213ensymd 7604 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~~  ~P B )
22 entr 7605 . . . 4  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )
2320, 21, 22syl2anc 659 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ~P B )
24 domentr 7612 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  ( ( 2o 
^m  B )  ^m  B )  /\  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )  -> 
( A  ^m  B
)  ~<_  ~P B )
257, 23, 24syl2anc 659 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 mapdom1 7720 . . . 4  |-  ( 2o  ~<_  A  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
2726ad2antrl 726 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
28 endomtr 7611 . . 3  |-  ( ( ~P B  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~<_  ( A  ^m  B
) )  ->  ~P B  ~<_  ( A  ^m  B ) )
293, 27, 28syl2anc 659 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~<_  ( A  ^m  B ) )
30 sbth 7675 . 2  |-  ( ( ( A  ^m  B
)  ~<_  ~P B  /\  ~P B  ~<_  ( A  ^m  B ) )  -> 
( A  ^m  B
)  ~~  ~P B
)
3125, 29, 30syl2anc 659 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 367    e. wcel 1842   ~Pcpw 3955   class class class wbr 4395    X. cxp 4821   dom cdm 4823   Oncon0 5410  (class class class)co 6278   omcom 6683   2oc2o 7161    ^m cmap 7457    ~~ cen 7551    ~<_ cdom 7552   cardccrd 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-oi 7969  df-card 8352
This theorem is referenced by:  alephexp1  8986  hauspwdom  20294
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