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Theorem hsmexlem1 8807
Description: Lemma for hsmex 8813. 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 7997 . . 3  |-  Ord  dom  O
3 relwdom 8034 . . . . . . . 8  |-  Rel  ~<_*
43brrelexi 4837 . . . . . . 7  |-  ( A  ~<_*  B  ->  A  e.  _V )
54adantl 467 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  e.  _V )
6 uniexg 6546 . . . . . 6  |-  ( A  e.  _V  ->  U. A  e.  _V )
7 sucexg 6595 . . . . . 6  |-  ( U. A  e.  _V  ->  suc  U. A  e.  _V )
85, 6, 73syl 18 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  suc  U. A  e.  _V )
91oif 7998 . . . . . . 7  |-  O : dom  O --> A
10 onsucuni 6613 . . . . . . . 8  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
1110adantr 466 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  suc  U. A )
12 fss 5697 . . . . . . 7  |-  ( ( O : dom  O --> A  /\  A  C_  suc  U. A )  ->  O : dom  O --> suc  U. A )
139, 11, 12sylancr 667 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O --> suc  U. A )
141oismo 8008 . . . . . . . 8  |-  ( A 
C_  On  ->  ( Smo 
O  /\  ran  O  =  A ) )
1514adantr 466 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( Smo  O  /\  ran  O  =  A ) )
1615simpld 460 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Smo  O )
17 ssorduni 6570 . . . . . . . 8  |-  ( A 
C_  On  ->  Ord  U. A )
1817adantr 466 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  U. A
)
19 ordsuc 6599 . . . . . . 7  |-  ( Ord  U. A  <->  Ord  suc  U. A )
2018, 19sylib 199 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  Ord  suc  U. A )
21 smorndom 7042 . . . . . 6  |-  ( ( O : dom  O --> suc  U. A  /\  Smo  O  /\  Ord  suc  U. A )  ->  dom  O 
C_  suc  U. A )
2213, 16, 20, 21syl3anc 1264 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  C_  suc  U. A )
238, 22ssexd 4514 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e. 
_V )
24 elong 5393 . . . 4  |-  ( dom 
O  e.  _V  ->  ( dom  O  e.  On  <->  Ord 
dom  O ) )
2523, 24syl 17 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( dom  O  e.  On  <->  Ord  dom  O
) )
262, 25mpbiri 236 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  On )
27 canth2g 7679 . . . 4  |-  ( dom 
O  e.  _V  ->  dom 
O  ~<  ~P dom  O
)
28 sdomdom 7551 . . . 4  |-  ( dom 
O  ~<  ~P dom  O  ->  dom  O  ~<_  ~P dom  O )
2923, 27, 283syl 18 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P
dom  O )
30 simpl 458 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  A  C_  On )
31 epweon 6568 . . . . . . . . . . 11  |-  _E  We  On
32 wess 4783 . . . . . . . . . . 11  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
3330, 31, 32mpisyl 21 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  _E  We  A )
34 epse 4779 . . . . . . . . . 10  |-  _E Se  A
351oiiso2 7999 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A )  ->  O  Isom  _E  ,  _E  ( dom  O ,  ran  O
) )
3633, 34, 35sylancl 666 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O  Isom  _E  ,  _E  ( dom 
O ,  ran  O
) )
37 isof1o 6175 . . . . . . . . 9  |-  ( O 
Isom  _E  ,  _E  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
3836, 37syl 17 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> ran  O )
3915simprd 464 . . . . . . . . 9  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ran  O  =  A )
40 f1oeq3 5767 . . . . . . . . 9  |-  ( ran 
O  =  A  -> 
( O : dom  O -1-1-onto-> ran 
O  <->  O : dom  O -1-1-onto-> A
) )
4139, 40syl 17 . . . . . . . 8  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ( O : dom  O -1-1-onto-> ran  O  <->  O : dom  O -1-1-onto-> A ) )
4238, 41mpbid 213 . . . . . . 7  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  O : dom  O -1-1-onto-> A )
43 f1oen2g 7540 . . . . . . 7  |-  ( ( dom  O  e.  On  /\  A  e.  _V  /\  O : dom  O -1-1-onto-> A )  ->  dom  O  ~~  A )
4426, 5, 42, 43syl3anc 1264 . . . . . 6  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~~  A )
45 endom 7550 . . . . . 6  |-  ( dom 
O  ~~  A  ->  dom 
O  ~<_  A )
46 domwdom 8042 . . . . . 6  |-  ( dom 
O  ~<_  A  ->  dom  O  ~<_*  A )
4744, 45, 463syl 18 . . . . 5  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  A
)
48 wdomtr 8043 . . . . 5  |-  ( ( dom  O  ~<_*  A  /\  A  ~<_*  B
)  ->  dom  O  ~<_*  B
)
4947, 48sylancom 671 . . . 4  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_*  B
)
50 wdompwdom 8046 . . . 4  |-  ( dom 
O  ~<_*  B  ->  ~P dom  O  ~<_  ~P B )
5149, 50syl 17 . . 3  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  ~P dom  O  ~<_  ~P B )
52 domtr 7576 . . 3  |-  ( ( dom  O  ~<_  ~P dom  O  /\  ~P dom  O  ~<_  ~P B )  ->  dom  O  ~<_  ~P B )
5329, 51, 52syl2anc 665 . 2  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  ~<_  ~P B )
54 elharval 8031 . 2  |-  ( dom 
O  e.  (har `  ~P B )  <->  ( dom  O  e.  On  /\  dom  O  ~<_  ~P B ) )
5526, 53, 54sylanbrc 668 1  |-  ( ( A  C_  On  /\  A  ~<_*  B )  ->  dom  O  e.  (har `  ~P B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3022    C_ wss 3379   ~Pcpw 3924   U.cuni 4162   class class class wbr 4366    _E cep 4705   Se wse 4753    We wwe 4754   dom cdm 4796   ran crn 4797   Ord word 5384   Oncon0 5385   suc csuc 5387   -->wf 5540   -1-1-onto->wf1o 5543   ` cfv 5544    Isom wiso 5545   Smo wsmo 7019    ~~ cen 7521    ~<_ cdom 7522    ~< csdm 7523  OrdIsocoi 7977  harchar 8024    ~<_* cwdom 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-wrecs 6983  df-smo 7020  df-recs 7045  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-oi 7978  df-har 8026  df-wdom 8027
This theorem is referenced by:  hsmexlem2  8808  hsmexlem4  8810
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