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Theorem hauspwdom 17227
Description: Simplify the cardinal  A ^ NN of hausmapdom 17226 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 17226 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
32adantr 451 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN ) )
4 simprr 733 . . . 4  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  NN 
~<_  B )
5 1nn 9757 . . . . 5  |-  1  e.  NN
6 noel 3459 . . . . . . 7  |-  -.  1  e.  (/)
7 eleq2 2344 . . . . . . 7  |-  ( NN  =  (/)  ->  ( 1  e.  NN  <->  1  e.  (/) ) )
86, 7mtbiri 294 . . . . . 6  |-  ( NN  =  (/)  ->  -.  1  e.  NN )
98adantr 451 . . . . 5  |-  ( ( NN  =  (/)  /\  A  =  (/) )  ->  -.  1  e.  NN )
105, 9mt2 170 . . . 4  |-  -.  ( NN  =  (/)  /\  A  =  (/) )
11 mapdom2 7032 . . . 4  |-  ( ( NN  ~<_  B  /\  -.  ( NN  =  (/)  /\  A  =  (/) ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
124, 10, 11sylancl 643 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
13 sdomdom 6889 . . . . . . 7  |-  ( A 
~<  2o  ->  A  ~<_  2o )
1413adantl 452 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  A  ~<_  2o )
15 mapdom1 7026 . . . . . 6  |-  ( A  ~<_  2o  ->  ( A  ^m  B )  ~<_  ( 2o 
^m  B ) )
1614, 15syl 15 . . . . 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 6869 . . . . . . . . 9  |-  Rel  ~<_
1817brrelex2i 4730 . . . . . . . 8  |-  ( NN  ~<_  B  ->  B  e.  _V )
1918ad2antll 709 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  _V )
20 pw2eng 6968 . . . . . . 7  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
21 ensym 6910 . . . . . . 7  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2219, 20, 213syl 18 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2322adantr 451 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( 2o  ^m  B )  ~~  ~P B )
24 domentr 6920 . . . . 5  |-  ( ( ( A  ^m  B
)  ~<_  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( A  ^m  B )  ~<_  ~P B )
2516, 23, 24syl2anc 642 . . . 4  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 onfin2 7052 . . . . . . . . 9  |-  om  =  ( On  i^i  Fin )
27 inss2 3390 . . . . . . . . 9  |-  ( On 
i^i  Fin )  C_  Fin
2826, 27eqsstri 3208 . . . . . . . 8  |-  om  C_  Fin
29 2onn 6638 . . . . . . . 8  |-  2o  e.  om
3028, 29sselii 3177 . . . . . . 7  |-  2o  e.  Fin
31 simprl 732 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  ~<_  ~P B )
3217brrelexi 4729 . . . . . . . 8  |-  ( A  ~<_  ~P B  ->  A  e.  _V )
3331, 32syl 15 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  e.  _V )
34 fidomtri 7626 . . . . . . 7  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3530, 33, 34sylancr 644 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3635biimpar 471 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  -.  A  ~<  2o )  ->  2o  ~<_  A )
37 numth3 8097 . . . . . . . . 9  |-  ( B  e.  _V  ->  B  e.  dom  card )
3819, 37syl 15 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  dom  card )
3938adantr 451 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  B  e.  dom  card )
40 nnenom 11042 . . . . . . . . . 10  |-  NN  ~~  om
4140ensymi 6911 . . . . . . . . 9  |-  om  ~~  NN
42 endomtr 6919 . . . . . . . . 9  |-  ( ( om  ~~  NN  /\  NN 
~<_  B )  ->  om  ~<_  B )
4341, 4, 42sylancr 644 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  om 
~<_  B )
4443adantr 451 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  om  ~<_  B )
45 simpr 447 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  2o  ~<_  A )
4631adantr 451 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  A  ~<_  ~P B
)
47 mappwen 7739 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
4839, 44, 45, 46, 47syl22anc 1183 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~~  ~P B )
49 endom 6888 . . . . . 6  |-  ( ( A  ^m  B ) 
~~  ~P B  ->  ( A  ^m  B )  ~<_  ~P B )
5048, 49syl 15 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~<_  ~P B
)
5136, 50syldan 456 . . . 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 766 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  B
)  ~<_  ~P B )
53 domtr 6914 . . 3  |-  ( ( ( A  ^m  NN )  ~<_  ( A  ^m  B )  /\  ( A  ^m  B )  ~<_  ~P B )  ->  ( A  ^m  NN )  ~<_  ~P B )
5412, 52, 53syl2anc 642 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ~P B )
55 domtr 6914 . 2  |-  ( ( ( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN )  /\  ( A  ^m  NN )  ~<_  ~P B
)  ->  ( ( cls `  J ) `  A )  ~<_  ~P B
)
563, 54, 55syl2anc 642 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   cardccrd 7568   1c1 8738   NNcn 9746   clsccl 16755   Hauscha 17036   1stcc1stc 17163
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  ax-cc 8061  ax-ac2 8089  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  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-iin 3908  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-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-acn 7575  df-ac 7743  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-top 16636  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-lm 16959  df-haus 17043  df-1stc 17165
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