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Theorem hauspwdom 19796
Description: Simplify the cardinal  A ^ NN of hausmapdom 19795 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 19795 . . 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 10547 . . . . 5  |-  1  e.  NN
6 noel 3789 . . . . . . 7  |-  -.  1  e.  (/)
7 eleq2 2540 . . . . . . 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 7688 . . . 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 7543 . . . . . . 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 7682 . . . . . 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 7522 . . . . . . . . 9  |-  Rel  ~<_
1817brrelex2i 5041 . . . . . . . 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 7623 . . . . . . 7  |-  ( B  e.  _V  ->  ~P B  ~~  ( 2o  ^m  B ) )
21 ensym 7564 . . . . . . 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 7574 . . . . 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 7709 . . . . . . . . 9  |-  om  =  ( On  i^i  Fin )
27 inss2 3719 . . . . . . . . 9  |-  ( On 
i^i  Fin )  C_  Fin
2826, 27eqsstri 3534 . . . . . . . 8  |-  om  C_  Fin
29 2onn 7289 . . . . . . . 8  |-  2o  e.  om
3028, 29sselii 3501 . . . . . . 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 5040 . . . . . . . 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 8374 . . . . . . 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 8850 . . . . . . . . 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 12058 . . . . . . . . . 10  |-  NN  ~~  om
4140ensymi 7565 . . . . . . . . 9  |-  om  ~~  NN
42 endomtr 7573 . . . . . . . . 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 8493 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( 2o  ~<_  A  /\  A  ~<_  ~P B ) )  ->  ( A  ^m  B )  ~~  ~P B )
4839, 44, 45, 46, 47syl22anc 1229 . . . . . 6  |-  ( ( ( ( J  e. 
Haus  /\  J  e.  1stc  /\  A  C_  X )  /\  ( A  ~<_  ~P B  /\  NN  ~<_  B ) )  /\  2o  ~<_  A )  ->  ( A  ^m  B )  ~~  ~P B )
49 endom 7542 . . . . . 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 7568 . . 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 7568 . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   Oncon0 4878   dom cdm 4999   ` cfv 5588  (class class class)co 6284   omcom 6684   2oc2o 7124    ^m cmap 7420    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516   cardccrd 8316   1c1 9493   NNcn 10536   clsccl 19313   Hauscha 19603   1stcc1stc 19732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-ac2 8843  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-card 8320  df-acn 8323  df-ac 8497  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-top 19194  df-topon 19197  df-cld 19314  df-ntr 19315  df-cls 19316  df-lm 19524  df-haus 19610  df-1stc 19734
This theorem is referenced by: (None)
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