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Theorem hsmexlem1 8239
Description: Lemma for hsmex 8245. 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 7431 . . 3  |-  Ord  dom  O
3 relwdom 7467 . . . . . . . 8  |-  Rel  ~<_*
43brrelexi 4858 . . . . . . 7  |-  ( A  ~<_*  B  ->  A  e.  _V )
54adantl 453 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  e.  _V )
6 uniexg 4646 . . . . . 6  |-  ( A  e.  _V  ->  U. A  e.  _V )
7 sucexg 4730 . . . . . 6  |-  ( U. A  e.  _V  ->  suc  U. A  e.  _V )
85, 6, 73syl 19 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  suc  U. A  e.  _V )
91oif 7432 . . . . . . 7  |-  O : dom  O --> A
10 onsucuni 4748 . . . . . . . 8  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
1110adantr 452 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  suc  U. A )
12 fss 5539 . . . . . . 7  |-  ( ( O : dom  O --> A  /\  A  C_  suc  U. A )  ->  O : dom  O --> suc  U. A )
139, 11, 12sylancr 645 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O --> suc  U. A )
141oismo 7442 . . . . . . . 8  |-  ( A 
C_  On  ->  ( Smo 
O  /\  ran  O  =  A ) )
1514adantr 452 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( Smo  O  /\  ran  O  =  A ) )
1615simpld 446 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Smo  O )
17 ssorduni 4706 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
1817adantr 452 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  U. A
)
19 ordsuc 4734 . . . . . . 7  |-  ( Ord  U. A  <->  Ord  suc  U. A )
2018, 19sylib 189 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  suc  U. A )
21 smorndom 6566 . . . . . 6  |-  ( ( O : dom  O --> suc  U. A  /\  Smo  O  /\  Ord  suc  U. A )  ->  dom  O 
C_  suc  U. A )
2213, 16, 20, 21syl3anc 1184 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  C_  suc  U. A )
238, 22ssexd 4291 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e. 
_V )
24 elong 4530 . . . 4  |-  ( dom 
O  e.  _V  ->  ( dom  O  e.  On  <->  Ord 
dom  O ) )
2523, 24syl 16 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( dom  O  e.  On  <->  Ord  dom  O
) )
262, 25mpbiri 225 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  On )
27 canth2g 7197 . . . 4  |-  ( dom 
O  e.  _V  ->  dom 
O  ~<  ~P dom  O
)
28 sdomdom 7071 . . . 4  |-  ( dom 
O  ~<  ~P dom  O  ->  dom  O  ~<_  ~P dom  O )
2923, 27, 283syl 19 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P
dom  O )
30 simpl 444 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  On )
31 epweon 4704 . . . . . . . . . . 11  |-  _E  We  On
32 wess 4510 . . . . . . . . . . 11  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
3330, 31, 32ee10 1382 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  _E  We  A )
34 epse 4506 . . . . . . . . . 10  |-  _E Se  A
351oiiso2 7433 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A )  ->  O  Isom  _E  ,  _E  ( dom  O ,  ran  O
) )
3633, 34, 35sylancl 644 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ran  O
) )
37 isof1o 5984 . . . . . . . . 9  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
3836, 37syl 16 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> ran  O )
3915simprd 450 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ran  O  =  A )
40 f1oeq3 5607 . . . . . . . . 9  |-  ( ran 
O  =  A  -> 
( O : dom  O -1-1-onto-> ran 
O  <->  O : dom  O -1-1-onto-> A
) )
4139, 40syl 16 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( O : dom  O -1-1-onto-> ran  O  <->  O : dom  O -1-1-onto-> A ) )
4238, 41mpbid 202 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> A )
43 f1oen2g 7060 . . . . . . 7  |-  ( ( dom  O  e.  On  /\  A  e.  _V  /\  O : dom  O -1-1-onto-> A )  ->  dom  O  ~~  A )
4426, 5, 42, 43syl3anc 1184 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~~  A )
45 endom 7070 . . . . . 6  |-  ( dom 
O  ~~  A  ->  dom 
O  ~<_  A )
46 domwdom 7475 . . . . . 6  |-  ( dom 
O  ~<_  A  ->  dom  O  ~<_*  A )
4744, 45, 463syl 19 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  A
)
48 wdomtr 7476 . . . . 5  |-  ( ( dom  O  ~<_*  A  /\  A  ~<_*  B
)  ->  dom  O  ~<_*  B
)
4947, 48sylancom 649 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  B
)
50 wdompwdom 7479 . . . 4  |-  ( dom 
O  ~<_*  B  ->  ~P dom  O  ~<_  ~P B )
5149, 50syl 16 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ~P dom  O  ~<_  ~P B )
52 domtr 7096 . . 3  |-  ( ( dom  O  ~<_  ~P dom  O  /\  ~P dom  O  ~<_  ~P B )  ->  dom  O  ~<_  ~P B )
5329, 51, 52syl2anc 643 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P B )
54 elharval 7464 . 2  |-  ( dom 
O  e.  (har `  ~P B )  <->  ( dom  O  e.  On  /\  dom  O  ~<_  ~P B ) )
5526, 53, 54sylanbrc 646 1  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   class class class wbr 4153    _E cep 4433   Se wse 4480    We wwe 4481   Ord word 4521   Oncon0 4522   suc csuc 4524   dom cdm 4818   ran crn 4819   -->wf 5390   -1-1-onto->wf1o 5393   ` cfv 5394    Isom wiso 5395   Smo wsmo 6543    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044  OrdIsocoi 7411  harchar 7457    ~<_* cwdom 7458
This theorem is referenced by:  hsmexlem2  8240  hsmexlem4  8242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-riota 6485  df-smo 6544  df-recs 6569  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-oi 7412  df-har 7459  df-wdom 7460
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