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Theorem mappwen 7672
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 736 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ~P B
)
2 pw2eng 6901 . . . . . 6  |-  ( B  e.  dom  card  ->  ~P B  ~~  ( 2o 
^m  B ) )
32ad2antrr 709 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~~  ( 2o  ^m  B ) )
4 domentr 6853 . . . . 5  |-  ( ( A  ~<_  ~P B  /\  ~P B  ~~  ( 2o  ^m  B ) )  ->  A  ~<_  ( 2o  ^m  B ) )
51, 3, 4syl2anc 645 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  A  ~<_  ( 2o 
^m  B ) )
6 mapdom1 6959 . . . 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 6420 . . . . . . 7  |-  2o  e.  On
98a1i 12 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  2o  e.  On )
10 simpll 733 . . . . . 6  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  B  e.  dom  card )
11 mapxpen 6960 . . . . . 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 1187 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  ( B  X.  B ) ) )
138elexi 2749 . . . . . . 7  |-  2o  e.  _V
1413enref 6827 . . . . . 6  |-  2o  ~~  2o
15 infxpidm2 7577 . . . . . . 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 6958 . . . . . 6  |-  ( ( 2o  ~~  2o  /\  ( B  X.  B
)  ~~  B )  ->  ( 2o  ^m  ( B  X.  B ) ) 
~~  ( 2o  ^m  B ) )
1814, 16, 17sylancr 647 . . . . 5  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  ( B  X.  B
) )  ~~  ( 2o  ^m  B ) )
19 entr 6846 . . . . 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 645 . . . 4  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ( 2o  ^m  B ) )
21 ensym 6843 . . . . 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 6846 . . . 4  |-  ( ( ( ( 2o  ^m  B )  ^m  B
)  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )
2420, 22, 23syl2anc 645 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( ( 2o 
^m  B )  ^m  B )  ~~  ~P B )
25 domentr 6853 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  ( ( 2o 
^m  B )  ^m  B )  /\  (
( 2o  ^m  B
)  ^m  B )  ~~  ~P B )  -> 
( A  ^m  B
)  ~<_  ~P B )
267, 24, 25syl2anc 645 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~<_  ~P B
)
27 mapdom1 6959 . . . 4  |-  ( 2o  ~<_  A  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
2827ad2antrl 711 . . 3  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( 2o  ^m  B )  ~<_  ( A  ^m  B ) )
29 endomtr 6852 . . 3  |-  ( ( ~P B  ~~  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~<_  ( A  ^m  B
) )  ->  ~P B  ~<_  ( A  ^m  B ) )
303, 28, 29syl2anc 645 . 2  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ~P B  ~<_  ( A  ^m  B ) )
31 sbth 6914 . 2  |-  ( ( ( A  ^m  B
)  ~<_  ~P B  /\  ~P B  ~<_  ( A  ^m  B ) )  -> 
( A  ^m  B
)  ~~  ~P B
)
3226, 30, 31syl2anc 645 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 1621   ~Pcpw 3566   class class class wbr 3963   Oncon0 4329   omcom 4593    X. cxp 4624   dom cdm 4626  (class class class)co 5757   2oc2o 6406    ^m cmap 6705    ~~ cen 6793    ~<_ cdom 6794   cardccrd 7501
This theorem is referenced by:  alephexp1  8134  hauspwdom  17154
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-card 7505
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