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Theorem hsmexlem1 8068
Description: Lemma for hsmex 8074. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
hsmexlem.o  |-  O  = OrdIso
(  _E  ,  A
)
Assertion
Ref Expression
hsmexlem1  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )

Proof of Theorem hsmexlem1
StepHypRef Expression
1 hsmexlem.o . . . 4  |-  O  = OrdIso
(  _E  ,  A
)
21oicl 7260 . . 3  |-  Ord  dom  O
31oif 7261 . . . . . . 7  |-  O : dom  O --> A
4 onsucuni 4635 . . . . . . . 8  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
54adantr 451 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  suc  U. A )
6 fss 5413 . . . . . . 7  |-  ( ( O : dom  O --> A  /\  A  C_  suc  U. A )  ->  O : dom  O --> suc  U. A )
73, 5, 6sylancr 644 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O --> suc  U. A )
81oismo 7271 . . . . . . . 8  |-  ( A 
C_  On  ->  ( Smo 
O  /\  ran  O  =  A ) )
98adantr 451 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( Smo  O  /\  ran  O  =  A ) )
109simpld 445 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Smo  O )
11 ssorduni 4593 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
1211adantr 451 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  U. A
)
13 ordsuc 4621 . . . . . . 7  |-  ( Ord  U. A  <->  Ord  suc  U. A )
1412, 13sylib 188 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  suc  U. A )
15 smorndom 6401 . . . . . 6  |-  ( ( O : dom  O --> suc  U. A  /\  Smo  O  /\  Ord  suc  U. A )  ->  dom  O 
C_  suc  U. A )
167, 10, 14, 15syl3anc 1182 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  C_  suc  U. A )
17 relwdom 7296 . . . . . . . 8  |-  Rel  ~<_*
1817brrelexi 4745 . . . . . . 7  |-  ( A  ~<_*  B  ->  A  e.  _V )
1918adantl 452 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  e.  _V )
20 uniexg 4533 . . . . . 6  |-  ( A  e.  _V  ->  U. A  e.  _V )
21 sucexg 4617 . . . . . 6  |-  ( U. A  e.  _V  ->  suc  U. A  e.  _V )
2219, 20, 213syl 18 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  suc  U. A  e.  _V )
23 ssexg 4176 . . . . 5  |-  ( ( dom  O  C_  suc  U. A  /\  suc  U. A  e.  _V )  ->  dom  O  e.  _V )
2416, 22, 23syl2anc 642 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e. 
_V )
25 elong 4416 . . . 4  |-  ( dom 
O  e.  _V  ->  ( dom  O  e.  On  <->  Ord 
dom  O ) )
2624, 25syl 15 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( dom  O  e.  On  <->  Ord  dom  O
) )
272, 26mpbiri 224 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  On )
28 canth2g 7031 . . . 4  |-  ( dom 
O  e.  _V  ->  dom 
O  ~<  ~P dom  O
)
29 sdomdom 6905 . . . 4  |-  ( dom 
O  ~<  ~P dom  O  ->  dom  O  ~<_  ~P dom  O )
3024, 28, 293syl 18 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P
dom  O )
31 simpl 443 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  On )
32 epweon 4591 . . . . . . . . . . 11  |-  _E  We  On
33 wess 4396 . . . . . . . . . . 11  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
3431, 32, 33ee10 1366 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  _E  We  A )
35 epse 4392 . . . . . . . . . 10  |-  _E Se  A
361oiiso2 7262 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A )  ->  O  Isom  _E  ,  _E  ( dom  O ,  ran  O
) )
3734, 35, 36sylancl 643 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ran  O
) )
38 isof1o 5838 . . . . . . . . 9  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
3937, 38syl 15 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> ran  O )
409simprd 449 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ran  O  =  A )
41 f1oeq3 5481 . . . . . . . . 9  |-  ( ran 
O  =  A  -> 
( O : dom  O -1-1-onto-> ran 
O  <->  O : dom  O -1-1-onto-> A
) )
4240, 41syl 15 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( O : dom  O -1-1-onto-> ran  O  <->  O : dom  O -1-1-onto-> A ) )
4339, 42mpbid 201 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> A )
44 f1oen2g 6894 . . . . . . 7  |-  ( ( dom  O  e.  On  /\  A  e.  _V  /\  O : dom  O -1-1-onto-> A )  ->  dom  O  ~~  A )
4527, 19, 43, 44syl3anc 1182 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~~  A )
46 endom 6904 . . . . . 6  |-  ( dom 
O  ~~  A  ->  dom 
O  ~<_  A )
47 domwdom 7304 . . . . . 6  |-  ( dom 
O  ~<_  A  ->  dom  O  ~<_*  A )
4845, 46, 473syl 18 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  A
)
49 wdomtr 7305 . . . . 5  |-  ( ( dom  O  ~<_*  A  /\  A  ~<_*  B
)  ->  dom  O  ~<_*  B
)
5048, 49sylancom 648 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  B
)
51 wdompwdom 7308 . . . 4  |-  ( dom 
O  ~<_*  B  ->  ~P dom  O  ~<_  ~P B )
5250, 51syl 15 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ~P dom  O  ~<_  ~P B )
53 domtr 6930 . . 3  |-  ( ( dom  O  ~<_  ~P dom  O  /\  ~P dom  O  ~<_  ~P B )  ->  dom  O  ~<_  ~P B )
5430, 52, 53syl2anc 642 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P B )
55 elharval 7293 . 2  |-  ( dom 
O  e.  (har `  ~P B )  <->  ( dom  O  e.  On  /\  dom  O  ~<_  ~P B ) )
5627, 54, 55sylanbrc 645 1  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    _E cep 4319   Se wse 4366    We wwe 4367   Ord word 4407   Oncon0 4408   suc csuc 4410   dom cdm 4705   ran crn 4706   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   Smo wsmo 6378    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878  OrdIsocoi 7240  harchar 7286    ~<_* cwdom 7287
This theorem is referenced by:  hsmexlem2  8069  hsmexlem4  8071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-smo 6379  df-recs 6404  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-wdom 7289
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