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Theorem oieu 7960
Description: Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1  |-  F  = OrdIso
( R ,  A
)
Assertion
Ref Expression
oieu  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  <->  ( B  =  dom  F  /\  G  =  F ) ) )

Proof of Theorem oieu
StepHypRef Expression
1 simprr 756 . . . . . 6  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  Isom  _E  ,  R  ( B ,  A ) )
2 oicl.1 . . . . . . . . 9  |-  F  = OrdIso
( R ,  A
)
32ordtype 7953 . . . . . . . 8  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
43adantr 465 . . . . . . 7  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
5 isocnv 6212 . . . . . . 7  |-  ( F 
Isom  _E  ,  R  ( dom  F ,  A
)  ->  `' F  Isom  R ,  _E  ( A ,  dom  F ) )
64, 5syl 16 . . . . . 6  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  `' F  Isom  R ,  _E  ( A ,  dom  F
) )
7 isotr 6218 . . . . . 6  |-  ( ( G  Isom  _E  ,  R  ( B ,  A )  /\  `' F  Isom  R ,  _E  ( A ,  dom  F ) )  ->  ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F ) )
81, 6, 7syl2anc 661 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( `' F  o.  G
)  Isom  _E  ,  _E  ( B ,  dom  F
) )
9 simprl 755 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  B )
102oicl 7950 . . . . . 6  |-  Ord  dom  F
1110a1i 11 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  dom 
F )
12 ordiso2 7936 . . . . 5  |-  ( ( ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F )  /\  Ord  B  /\  Ord  dom  F )  ->  B  =  dom  F
)
138, 9, 11, 12syl3anc 1228 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  B  =  dom  F )
14 ordwe 4891 . . . . . 6  |-  ( Ord 
B  ->  _E  We  B )
1514ad2antrl 727 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E  We  B )
16 epse 4862 . . . . . 6  |-  _E Se  B
1716a1i 11 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E Se  B )
18 isoeq4 6204 . . . . . . 7  |-  ( B  =  dom  F  -> 
( F  Isom  _E  ,  R  ( B ,  A )  <->  F  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
1913, 18syl 16 . . . . . 6  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( F  Isom  _E  ,  R  ( B ,  A )  <-> 
F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
204, 19mpbird 232 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( B ,  A ) )
21 weisoeq 6237 . . . . 5  |-  ( ( (  _E  We  B  /\  _E Se  B )  /\  ( G  Isom  _E  ,  R  ( B ,  A )  /\  F  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  =  F )
2215, 17, 1, 20, 21syl22anc 1229 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  =  F )
2313, 22jca 532 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( B  =  dom  F  /\  G  =  F )
)
2423ex 434 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  ->  ( B  =  dom  F  /\  G  =  F ) ) )
253, 10jctil 537 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  /\  F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
26 ordeq 4885 . . . . 5  |-  ( B  =  dom  F  -> 
( Ord  B  <->  Ord  dom  F
) )
2726adantr 465 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( Ord  B  <->  Ord  dom  F ) )
28 isoeq4 6204 . . . . 5  |-  ( B  =  dom  F  -> 
( G  Isom  _E  ,  R  ( B ,  A )  <->  G  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
29 isoeq1 6201 . . . . 5  |-  ( G  =  F  ->  ( G  Isom  _E  ,  R  ( dom  F ,  A
)  <->  F  Isom  _E  ,  R  ( dom  F ,  A ) ) )
3028, 29sylan9bb 699 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( G  Isom  _E  ,  R  ( B ,  A )  <-> 
F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
3127, 30anbi12d 710 . . 3  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  <-> 
( Ord  dom  F  /\  F  Isom  _E  ,  R  ( dom  F ,  A
) ) ) )
3225, 31syl5ibrcom 222 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( B  =  dom  F  /\  G  =  F )  ->  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) ) )
3324, 32impbid 191 1  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  <->  ( B  =  dom  F  /\  G  =  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    _E cep 4789   Se wse 4836    We wwe 4837   Ord word 4877   `'ccnv 4998   dom cdm 4999    o. ccom 5003    Isom wiso 5587  OrdIsocoi 7930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-recs 7039  df-oi 7931
This theorem is referenced by:  hartogslem1  7963  cantnfp1lem3  8095  oemapwe  8109  cantnffval2  8110  cantnfp1lem3OLD  8121  oemapweOLD  8131  cantnffval2OLD  8132  om2uzoi  12029
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