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Theorem hsmexlem1 8797
Description: Lemma for hsmex 8803. 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 7946 . . 3  |-  Ord  dom  O
3 relwdom 7984 . . . . . . . 8  |-  Rel  ~<_*
43brrelexi 5029 . . . . . . 7  |-  ( A  ~<_*  B  ->  A  e.  _V )
54adantl 464 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  e.  _V )
6 uniexg 6570 . . . . . 6  |-  ( A  e.  _V  ->  U. A  e.  _V )
7 sucexg 6618 . . . . . 6  |-  ( U. A  e.  _V  ->  suc  U. A  e.  _V )
85, 6, 73syl 20 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  suc  U. A  e.  _V )
91oif 7947 . . . . . . 7  |-  O : dom  O --> A
10 onsucuni 6636 . . . . . . . 8  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
1110adantr 463 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  suc  U. A )
12 fss 5721 . . . . . . 7  |-  ( ( O : dom  O --> A  /\  A  C_  suc  U. A )  ->  O : dom  O --> suc  U. A )
139, 11, 12sylancr 661 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O --> suc  U. A )
141oismo 7957 . . . . . . . 8  |-  ( A 
C_  On  ->  ( Smo 
O  /\  ran  O  =  A ) )
1514adantr 463 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( Smo  O  /\  ran  O  =  A ) )
1615simpld 457 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Smo  O )
17 ssorduni 6594 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
1817adantr 463 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  U. A
)
19 ordsuc 6622 . . . . . . 7  |-  ( Ord  U. A  <->  Ord  suc  U. A )
2018, 19sylib 196 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  suc  U. A )
21 smorndom 7031 . . . . . 6  |-  ( ( O : dom  O --> suc  U. A  /\  Smo  O  /\  Ord  suc  U. A )  ->  dom  O 
C_  suc  U. A )
2213, 16, 20, 21syl3anc 1226 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  C_  suc  U. A )
238, 22ssexd 4584 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e. 
_V )
24 elong 4875 . . . 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 233 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  On )
27 canth2g 7664 . . . 4  |-  ( dom 
O  e.  _V  ->  dom 
O  ~<  ~P dom  O
)
28 sdomdom 7536 . . . 4  |-  ( dom 
O  ~<  ~P dom  O  ->  dom  O  ~<_  ~P dom  O )
2923, 27, 283syl 20 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P
dom  O )
30 simpl 455 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  On )
31 epweon 6592 . . . . . . . . . . 11  |-  _E  We  On
32 wess 4855 . . . . . . . . . . 11  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
3330, 31, 32mpisyl 18 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  _E  We  A )
34 epse 4851 . . . . . . . . . 10  |-  _E Se  A
351oiiso2 7948 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A )  ->  O  Isom  _E  ,  _E  ( dom  O ,  ran  O
) )
3633, 34, 35sylancl 660 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ran  O
) )
37 isof1o 6196 . . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ran  O  =  A )
40 f1oeq3 5791 . . . . . . . . 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 210 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> A )
43 f1oen2g 7525 . . . . . . 7  |-  ( ( dom  O  e.  On  /\  A  e.  _V  /\  O : dom  O -1-1-onto-> A )  ->  dom  O  ~~  A )
4426, 5, 42, 43syl3anc 1226 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~~  A )
45 endom 7535 . . . . . 6  |-  ( dom 
O  ~~  A  ->  dom 
O  ~<_  A )
46 domwdom 7992 . . . . . 6  |-  ( dom 
O  ~<_  A  ->  dom  O  ~<_*  A )
4744, 45, 463syl 20 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  A
)
48 wdomtr 7993 . . . . 5  |-  ( ( dom  O  ~<_*  A  /\  A  ~<_*  B
)  ->  dom  O  ~<_*  B
)
4947, 48sylancom 665 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  B
)
50 wdompwdom 7996 . . . 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 7561 . . 3  |-  ( ( dom  O  ~<_  ~P dom  O  /\  ~P dom  O  ~<_  ~P B )  ->  dom  O  ~<_  ~P B )
5329, 51, 52syl2anc 659 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P B )
54 elharval 7981 . 2  |-  ( dom 
O  e.  (har `  ~P B )  <->  ( dom  O  e.  On  /\  dom  O  ~<_  ~P B ) )
5526, 53, 54sylanbrc 662 1  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   class class class wbr 4439    _E cep 4778   Se wse 4825    We wwe 4826   Ord word 4866   Oncon0 4867   suc csuc 4869   dom cdm 4988   ran crn 4989   -->wf 5566   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571   Smo wsmo 7008    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508  OrdIsocoi 7926  harchar 7974    ~<_* cwdom 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-smo 7009  df-recs 7034  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-oi 7927  df-har 7976  df-wdom 7977
This theorem is referenced by:  hsmexlem2  8798  hsmexlem4  8800
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