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Theorem hauspwdom 17564
Description: Simplify the cardinal  A ^ NN of hausmapdom 17563 to  ~P B  =  2 ^ B when  B is an infinite cardinal greater than  A. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypothesis
Ref Expression
hauspwdom.1  |-  X  = 
U. J
Assertion
Ref Expression
hauspwdom  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ~P B )

Proof of Theorem hauspwdom
StepHypRef Expression
1 hauspwdom.1 . . . 4  |-  X  = 
U. J
21hausmapdom 17563 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
32adantr 452 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN ) )
4 simprr 734 . . . 4  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  NN 
~<_  B )
5 1nn 10011 . . . . 5  |-  1  e.  NN
6 noel 3632 . . . . . . 7  |-  -.  1  e.  (/)
7 eleq2 2497 . . . . . . 7  |-  ( NN  =  (/)  ->  ( 1  e.  NN  <->  1  e.  (/) ) )
86, 7mtbiri 295 . . . . . 6  |-  ( NN  =  (/)  ->  -.  1  e.  NN )
98adantr 452 . . . . 5  |-  ( ( NN  =  (/)  /\  A  =  (/) )  ->  -.  1  e.  NN )
105, 9mt2 172 . . . 4  |-  -.  ( NN  =  (/)  /\  A  =  (/) )
11 mapdom2 7278 . . . 4  |-  ( ( NN  ~<_  B  /\  -.  ( NN  =  (/)  /\  A  =  (/) ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
124, 10, 11sylancl 644 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
13 sdomdom 7135 . . . . . . 7  |-  ( A 
~<  2o  ->  A  ~<_  2o )
1413adantl 453 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  A  ~<_  2o )
15 mapdom1 7272 . . . . . 6  |-  ( A  ~<_  2o  ->  ( A  ^m  B )  ~<_  ( 2o 
^m  B ) )
1614, 15syl 16 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ( 2o 
^m  B ) )
17 reldom 7115 . . . . . . . . 9  |-  Rel  ~<_
1817brrelex2i 4919 . . . . . . . 8  |-  ( NN  ~<_  B  ->  B  e.  _V )
1918ad2antll 710 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  _V )
20 pw2eng 7214 . . . . . . 7  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
21 ensym 7156 . . . . . . 7  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2219, 20, 213syl 19 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2322adantr 452 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( 2o  ^m  B )  ~~  ~P B )
24 domentr 7166 . . . . 5  |-  ( ( ( A  ^m  B
)  ~<_  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( A  ^m  B )  ~<_  ~P B )
2516, 23, 24syl2anc 643 . . . 4  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 onfin2 7298 . . . . . . . . 9  |-  om  =  ( On  i^i  Fin )
27 inss2 3562 . . . . . . . . 9  |-  ( On 
i^i  Fin )  C_  Fin
2826, 27eqsstri 3378 . . . . . . . 8  |-  om  C_  Fin
29 2onn 6883 . . . . . . . 8  |-  2o  e.  om
3028, 29sselii 3345 . . . . . . 7  |-  2o  e.  Fin
31 simprl 733 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  ~<_  ~P B )
3217brrelexi 4918 . . . . . . . 8  |-  ( A  ~<_  ~P B  ->  A  e.  _V )
3331, 32syl 16 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  e.  _V )
34 fidomtri 7880 . . . . . . 7  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3530, 33, 34sylancr 645 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3635biimpar 472 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  -.  A  ~<  2o )  ->  2o  ~<_  A )
37 numth3 8350 . . . . . . . . 9  |-  ( B  e.  _V  ->  B  e.  dom  card )
3819, 37syl 16 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  dom  card )
3938adantr 452 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  B  e.  dom  card )
40 nnenom 11319 . . . . . . . . . 10  |-  NN  ~~  om
4140ensymi 7157 . . . . . . . . 9  |-  om  ~~  NN
42 endomtr 7165 . . . . . . . . 9  |-  ( ( om  ~~  NN  /\  NN 
~<_  B )  ->  om  ~<_  B )
4341, 4, 42sylancr 645 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  om 
~<_  B )
4443adantr 452 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  om  ~<_  B )
45 simpr 448 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  2o  ~<_  A )
4631adantr 452 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  A  ~<_  ~P B
)
47 mappwen 7993 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
4839, 44, 45, 46, 47syl22anc 1185 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~~  ~P B )
49 endom 7134 . . . . . 6  |-  ( ( A  ^m  B ) 
~~  ~P B  ->  ( A  ^m  B )  ~<_  ~P B )
5048, 49syl 16 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~<_  ~P B
)
5136, 50syldan 457 . . . 4  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  -.  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ~P B
)
5225, 51pm2.61dan 767 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  B
)  ~<_  ~P B )
53 domtr 7160 . . 3  |-  ( ( ( A  ^m  NN )  ~<_  ( A  ^m  B )  /\  ( A  ^m  B )  ~<_  ~P B )  ->  ( A  ^m  NN )  ~<_  ~P B )
5412, 52, 53syl2anc 643 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ~P B )
55 domtr 7160 . 2  |-  ( ( ( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN )  /\  ( A  ^m  NN )  ~<_  ~P B
)  ->  ( ( cls `  J ) `  A )  ~<_  ~P B
)
563, 54, 55syl2anc 643 1  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ~P B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   U.cuni 4015   class class class wbr 4212   Oncon0 4581   omcom 4845   dom cdm 4878   ` cfv 5454  (class class class)co 6081   2oc2o 6718    ^m cmap 7018    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109   cardccrd 7822   1c1 8991   NNcn 10000   clsccl 17082   Hauscha 17372   1stcc1stc 17500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cc 8315  ax-ac2 8343  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-acn 7829  df-ac 7997  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-top 16963  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-lm 17293  df-haus 17379  df-1stc 17502
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