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Theorem mappwen 7735
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 735 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ~P B
)
2 pw2eng 6964 . . . . . 6  |-  ( B  e.  dom  card  ->  ~P B  ~~  ( 2o 
^m  B ) )
32ad2antrr 708 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~~  ( 2o  ^m  B ) )
4 domentr 6916 . . . . 5  |-  ( ( A  ~<_  ~P B  /\  ~P B  ~~  ( 2o  ^m  B ) )  ->  A  ~<_  ( 2o  ^m  B ) )
51, 3, 4syl2anc 644 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ( 2o 
^m  B ) )
6 mapdom1 7022 . . . 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 6483 . . . . . . 7  |-  2o  e.  On
98a1i 12 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  2o  e.  On )
10 simpll 732 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  B  e.  dom  card )
11 mapxpen 7023 . . . . . 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 1184 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
138elexi 2799 . . . . . . 7  |-  2o  e.  _V
1413enref 6890 . . . . . 6  |-  2o  ~~  2o
15 infxpidm2 7640 . . . . . . 7  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B )  -> 
( B  X.  B
)  ~~  B )
1615adantr 453 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( B  X.  B )  ~~  B
)
17 mapen 7021 . . . . . 6  |-  ( ( 2o  ~~  2o  /\  ( B  X.  B
)  ~~  B )  ->  ( 2o  ^m  ( B  X.  B ) ) 
~~  ( 2o  ^m  B ) )
1814, 16, 17sylancr 646 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  ( B  X.  B
) )  ~~  ( 2o  ^m  B ) )
19 entr 6909 . . . . 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 644 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
21 ensym 6906 . . . . 5  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
223, 21syl 17 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~~  ~P B )
23 entr 6909 . . . 4  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )
2420, 22, 23syl2anc 644 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ~P B )
25 domentr 6916 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  ( ( 2o 
^m  B )  ^m  B )  /\  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )  -> 
( A  ^m  B
)  ~<_  ~P B )
267, 24, 25syl2anc 644 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ~P B
)
27 mapdom1 7022 . . . 4  |-  ( 2o  ~<_  A  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
2827ad2antrl 710 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
29 endomtr 6915 . . 3  |-  ( ( ~P B  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~<_  ( A  ^m  B
) )  ->  ~P B  ~<_  ( A  ^m  B ) )
303, 28, 29syl2anc 644 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~<_  ( A  ^m  B ) )
31 sbth 6977 . 2  |-  ( ( ( A  ^m  B
)  ~<_  ~P B  /\  ~P B  ~<_  ( A  ^m  B ) )  -> 
( A  ^m  B
)  ~~  ~P B
)
3226, 30, 31syl2anc 644 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 6    /\ wa 360    e. wcel 1685   ~Pcpw 3627   class class class wbr 4025   Oncon0 4392   omcom 4656    X. cxp 4687   dom cdm 4689  (class class class)co 5820   2oc2o 6469    ^m cmap 6768    ~~ cen 6856    ~<_ cdom 6857   cardccrd 7564
This theorem is referenced by:  alephexp1  8197  hauspwdom  17222
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-card 7568
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