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Theorem hauspwdom 19104
Description: Simplify the cardinal  A ^ NN of hausmapdom 19103 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 19103 . . 3  |-  ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X
)  ->  ( ( cls `  J ) `  A )  ~<_  ( A  ^m  NN ) )
32adantr 465 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN ) )
4 simprr 756 . . . 4  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  NN 
~<_  B )
5 1nn 10332 . . . . 5  |-  1  e.  NN
6 noel 3640 . . . . . . 7  |-  -.  1  e.  (/)
7 eleq2 2503 . . . . . . 7  |-  ( NN  =  (/)  ->  ( 1  e.  NN  <->  1  e.  (/) ) )
86, 7mtbiri 303 . . . . . 6  |-  ( NN  =  (/)  ->  -.  1  e.  NN )
98adantr 465 . . . . 5  |-  ( ( NN  =  (/)  /\  A  =  (/) )  ->  -.  1  e.  NN )
105, 9mt2 179 . . . 4  |-  -.  ( NN  =  (/)  /\  A  =  (/) )
11 mapdom2 7481 . . . 4  |-  ( ( NN  ~<_  B  /\  -.  ( NN  =  (/)  /\  A  =  (/) ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
124, 10, 11sylancl 662 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ( A  ^m  B ) )
13 sdomdom 7336 . . . . . . 7  |-  ( A 
~<  2o  ->  A  ~<_  2o )
1413adantl 466 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  A  ~<_  2o )
15 mapdom1 7475 . . . . . 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 7315 . . . . . . . . 9  |-  Rel  ~<_
1817brrelex2i 4879 . . . . . . . 8  |-  ( NN  ~<_  B  ->  B  e.  _V )
1918ad2antll 728 . . . . . . 7  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  B  e.  _V )
20 pw2eng 7416 . . . . . . 7  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
21 ensym 7357 . . . . . . 7  |-  ( ~P B  ~~  ( 2o 
^m  B )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2219, 20, 213syl 20 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ^m  B
)  ~~  ~P B
)
2322adantr 465 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( 2o  ^m  B )  ~~  ~P B )
24 domentr 7367 . . . . 5  |-  ( ( ( A  ^m  B
)  ~<_  ( 2o  ^m  B )  /\  ( 2o  ^m  B )  ~~  ~P B )  ->  ( A  ^m  B )  ~<_  ~P B )
2516, 23, 24syl2anc 661 . . . 4  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  A  ~<  2o )  ->  ( A  ^m  B )  ~<_  ~P B
)
26 onfin2 7501 . . . . . . . . 9  |-  om  =  ( On  i^i  Fin )
27 inss2 3570 . . . . . . . . 9  |-  ( On 
i^i  Fin )  C_  Fin
2826, 27eqsstri 3385 . . . . . . . 8  |-  om  C_  Fin
29 2onn 7078 . . . . . . . 8  |-  2o  e.  om
3028, 29sselii 3352 . . . . . . 7  |-  2o  e.  Fin
31 simprl 755 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  A  ~<_  ~P B )
3217brrelexi 4878 . . . . . . . 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 8162 . . . . . . 7  |-  ( ( 2o  e.  Fin  /\  A  e.  _V )  ->  ( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3530, 33, 34sylancr 663 . . . . . 6  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( 2o  ~<_  A  <->  -.  A  ~<  2o ) )
3635biimpar 485 . . . . 5  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  -.  A  ~<  2o )  ->  2o  ~<_  A )
37 numth3 8638 . . . . . . . . 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 465 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  B  e.  dom  card )
40 nnenom 11801 . . . . . . . . . 10  |-  NN  ~~  om
4140ensymi 7358 . . . . . . . . 9  |-  om  ~~  NN
42 endomtr 7366 . . . . . . . . 9  |-  ( ( om  ~~  NN  /\  NN 
~<_  B )  ->  om  ~<_  B )
4341, 4, 42sylancr 663 . . . . . . . 8  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  ->  om 
~<_  B )
4443adantr 465 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  om  ~<_  B )
45 simpr 461 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  2o  ~<_  A )
4631adantr 465 . . . . . . 7  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  A  ~<_  ~P B
)
47 mappwen 8281 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
4839, 44, 45, 46, 47syl22anc 1219 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~~  ~P B )
49 endom 7335 . . . . . 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 470 . . . 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 789 . . 3  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  B
)  ~<_  ~P B )
53 domtr 7361 . . 3  |-  ( ( ( A  ^m  NN )  ~<_  ( A  ^m  B )  /\  ( A  ^m  B )  ~<_  ~P B )  ->  ( A  ^m  NN )  ~<_  ~P B )
5412, 52, 53syl2anc 661 . 2  |-  ( ( ( J  e.  Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  -> 
( A  ^m  NN )  ~<_  ~P B )
55 domtr 7361 . 2  |-  ( ( ( ( cls `  J
) `  A )  ~<_  ( A  ^m  NN )  /\  ( A  ^m  NN )  ~<_  ~P B
)  ->  ( ( cls `  J ) `  A )  ~<_  ~P B
)
563, 54, 55syl2anc 661 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2971    i^i cin 3326    C_ wss 3327   (/)c0 3636   ~Pcpw 3859   U.cuni 4090   class class class wbr 4291   Oncon0 4718   dom cdm 4839   ` cfv 5417  (class class class)co 6090   omcom 6475   2oc2o 6913    ^m cmap 7213    ~~ cen 7306    ~<_ cdom 7307    ~< csdm 7308   Fincfn 7309   cardccrd 8104   1c1 9282   NNcn 10321   clsccl 18621   Hauscha 18911   1stcc1stc 19040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cc 8603  ax-ac2 8631  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-oi 7723  df-card 8108  df-acn 8111  df-ac 8285  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-top 18502  df-topon 18505  df-cld 18622  df-ntr 18623  df-cls 18624  df-lm 18832  df-haus 18918  df-1stc 19042
This theorem is referenced by: (None)
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