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Theorem hsmexlem1 8052
Description: Lemma for hsmex 8058. 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 7244 . . 3  |-  Ord  dom  O
31oif 7245 . . . . . . 7  |-  O : dom  O --> A
4 onsucuni 4619 . . . . . . . 8  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
54adantr 451 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  suc  U. A )
6 fss 5397 . . . . . . 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 7255 . . . . . . . 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 4577 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
1211adantr 451 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  U. A
)
13 ordsuc 4605 . . . . . . 7  |-  ( Ord  U. A  <->  Ord  suc  U. A )
1412, 13sylib 188 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  suc  U. A )
15 smorndom 6385 . . . . . 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 7280 . . . . . . . 8  |-  Rel  ~<_*
1817brrelexi 4729 . . . . . . 7  |-  ( A  ~<_*  B  ->  A  e.  _V )
1918adantl 452 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  e.  _V )
20 uniexg 4517 . . . . . 6  |-  ( A  e.  _V  ->  U. A  e.  _V )
21 sucexg 4601 . . . . . 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 4160 . . . . 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 4400 . . . 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 7015 . . . 4  |-  ( dom 
O  e.  _V  ->  dom 
O  ~<  ~P dom  O
)
29 sdomdom 6889 . . . 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 4575 . . . . . . . . . . 11  |-  _E  We  On
33 wess 4380 . . . . . . . . . . 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 4376 . . . . . . . . . 10  |-  _E Se  A
361oiiso2 7246 . . . . . . . . . 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 5822 . . . . . . . . 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 5465 . . . . . . . . 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 6878 . . . . . . 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 6888 . . . . . 6  |-  ( dom 
O  ~~  A  ->  dom 
O  ~<_  A )
47 domwdom 7288 . . . . . 6  |-  ( dom 
O  ~<_  A  ->  dom  O  ~<_*  A )
4845, 46, 473syl 18 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  A
)
49 wdomtr 7289 . . . . 5  |-  ( ( dom  O  ~<_*  A  /\  A  ~<_*  B
)  ->  dom  O  ~<_*  B
)
5048, 49sylancom 648 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  B
)
51 wdompwdom 7292 . . . 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 6914 . . 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 7277 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023    _E cep 4303   Se wse 4350    We wwe 4351   Ord word 4391   Oncon0 4392   suc csuc 4394   dom cdm 4689   ran crn 4690   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   Smo wsmo 6362    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862  OrdIsocoi 7224  harchar 7270    ~<_* cwdom 7271
This theorem is referenced by:  hsmexlem2  8053  hsmexlem4  8055
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
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-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-iun 3907  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-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-riota 6304  df-smo 6363  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-oi 7225  df-har 7272  df-wdom 7273
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