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Theorem hauspwdom 20502
Description: Simplify the cardinal  A ^ NN of hausmapdom 20501 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 20501 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
32adantr 466 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN ) )
4 simprr 764 . . . 4  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  NN 
~<_  B )
5 1nn 10620 . . . . 5  |-  1  e.  NN
6 noel 3765 . . . . . . 7  |-  -.  1  e.  (/)
7 eleq2 2495 . . . . . . 7  |-  ( NN  =  (/)  ->  ( 1  e.  NN  <->  1  e.  (/) ) )
86, 7mtbiri 304 . . . . . 6  |-  ( NN  =  (/)  ->  -.  1  e.  NN )
98adantr 466 . . . . 5  |-  ( ( NN  =  (/)  /\  A  =  (/) )  ->  -.  1  e.  NN )
105, 9mt2 182 . . . 4  |-  -.  ( NN  =  (/)  /\  A  =  (/) )
11 mapdom2 7745 . . . 4  |-  ( ( NN  ~<_  B  /\  -.  ( NN  =  (/)  /\  A  =  (/) ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
124, 10, 11sylancl 666 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
13 sdomdom 7600 . . . . . . 7  |-  ( A 
~<  2o  ->  A  ~<_  2o )
1413adantl 467 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  A  ~<_  2o )
15 mapdom1 7739 . . . . . 6  |-  ( A  ~<_  2o  ->  ( A  ^m  B )  ~<_  ( 2o 
^m  B ) )
1614, 15syl 17 . . . . 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 7579 . . . . . . . . 9  |-  Rel  ~<_
1817brrelex2i 4891 . . . . . . . 8  |-  ( NN  ~<_  B  ->  B  e.  _V )
1918ad2antll 733 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  _V )
20 pw2eng 7680 . . . . . . 7  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
21 ensym 7621 . . . . . . 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 466 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( 2o  ^m  B )  ~~  ~P B )
24 domentr 7631 . . . . 5  |-  ( ( ( A  ^m  B
)  ~<_  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( A  ^m  B )  ~<_  ~P B )
2516, 23, 24syl2anc 665 . . . 4  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 onfin2 7766 . . . . . . . . 9  |-  om  =  ( On  i^i  Fin )
27 inss2 3683 . . . . . . . . 9  |-  ( On 
i^i  Fin )  C_  Fin
2826, 27eqsstri 3494 . . . . . . . 8  |-  om  C_  Fin
29 2onn 7345 . . . . . . . 8  |-  2o  e.  om
3028, 29sselii 3461 . . . . . . 7  |-  2o  e.  Fin
31 simprl 762 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  ~<_  ~P B )
3217brrelexi 4890 . . . . . . . 8  |-  ( A  ~<_  ~P B  ->  A  e.  _V )
3331, 32syl 17 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  e.  _V )
34 fidomtri 8428 . . . . . . 7  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3530, 33, 34sylancr 667 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3635biimpar 487 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  -.  A  ~<  2o )  ->  2o  ~<_  A )
37 numth3 8900 . . . . . . . . 9  |-  ( B  e.  _V  ->  B  e.  dom  card )
3819, 37syl 17 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  dom  card )
3938adantr 466 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  B  e.  dom  card )
40 nnenom 12192 . . . . . . . . . 10  |-  NN  ~~  om
4140ensymi 7622 . . . . . . . . 9  |-  om  ~~  NN
42 endomtr 7630 . . . . . . . . 9  |-  ( ( om  ~~  NN  /\  NN 
~<_  B )  ->  om  ~<_  B )
4341, 4, 42sylancr 667 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  om 
~<_  B )
4443adantr 466 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  om  ~<_  B )
45 simpr 462 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  2o  ~<_  A )
4631adantr 466 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  A  ~<_  ~P B
)
47 mappwen 8543 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
4839, 44, 45, 46, 47syl22anc 1265 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~~  ~P B )
49 endom 7599 . . . . . 6  |-  ( ( A  ^m  B ) 
~~  ~P B  ->  ( A  ^m  B )  ~<_  ~P B )
5048, 49syl 17 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~<_  ~P B
)
5136, 50syldan 472 . . . 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 798 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  B
)  ~<_  ~P B )
53 domtr 7625 . . 3  |-  ( ( ( A  ^m  NN )  ~<_  ( A  ^m  B )  /\  ( A  ^m  B )  ~<_  ~P B )  ->  ( A  ^m  NN )  ~<_  ~P B )
5412, 52, 53syl2anc 665 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ~P B )
55 domtr 7625 . 2  |-  ( ( ( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN )  /\  ( A  ^m  NN )  ~<_  ~P B
)  ->  ( ( cls `  J ) `  A )  ~<_  ~P B
)
563, 54, 55syl2anc 665 1  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081    i^i cin 3435    C_ wss 3436   (/)c0 3761   ~Pcpw 3979   U.cuni 4216   class class class wbr 4420   dom cdm 4849   Oncon0 5438   ` cfv 5597  (class class class)co 6301   omcom 6702   2oc2o 7180    ^m cmap 7476    ~~ cen 7570    ~<_ cdom 7571    ~< csdm 7572   Fincfn 7573   cardccrd 8370   1c1 9540   NNcn 10609   clsccl 20019   Hauscha 20310   1stcc1stc 20438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cc 8865  ax-ac2 8893  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-oi 8027  df-card 8374  df-acn 8377  df-ac 8547  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-top 19907  df-topon 19909  df-cld 20020  df-ntr 20021  df-cls 20022  df-lm 20231  df-haus 20317  df-1stc 20440
This theorem is referenced by: (None)
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