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Theorem oieu 7511
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 735 . . . . . 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 7504 . . . . . . . 8  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
43adantr 453 . . . . . . 7  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
5 isocnv 6053 . . . . . . 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 6059 . . . . . 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 644 . . . . 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 734 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  B )
102oicl 7501 . . . . . 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 7487 . . . . 5  |-  ( ( ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F )  /\  Ord  B  /\  Ord  dom  F )  ->  B  =  dom  F
)
138, 9, 11, 12syl3anc 1185 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  B  =  dom  F )
14 ordwe 4597 . . . . . 6  |-  ( Ord 
B  ->  _E  We  B )
1514ad2antrl 710 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E  We  B )
16 epse 4568 . . . . . 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 6045 . . . . . . 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 225 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( B ,  A ) )
21 weisoeq 6079 . . . . 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 1186 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  =  F )
2313, 22jca 520 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( B  =  dom  F  /\  G  =  F )
)
2423ex 425 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  ->  ( B  =  dom  F  /\  G  =  F ) ) )
253, 10jctil 525 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  /\  F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
26 ordeq 4591 . . . . 5  |-  ( B  =  dom  F  -> 
( Ord  B  <->  Ord  dom  F
) )
2726adantr 453 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( Ord  B  <->  Ord  dom  F ) )
28 isoeq4 6045 . . . . 5  |-  ( B  =  dom  F  -> 
( G  Isom  _E  ,  R  ( B ,  A )  <->  G  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
29 isoeq1 6042 . . . . 5  |-  ( G  =  F  ->  ( G  Isom  _E  ,  R  ( dom  F ,  A
)  <->  F  Isom  _E  ,  R  ( dom  F ,  A ) ) )
3028, 29sylan9bb 682 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( G  Isom  _E  ,  R  ( B ,  A )  <-> 
F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
3127, 30anbi12d 693 . . 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 215 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( B  =  dom  F  /\  G  =  F )  ->  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) ) )
3324, 32impbid 185 1  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  <->  ( B  =  dom  F  /\  G  =  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    _E cep 4495   Se wse 4542    We wwe 4543   Ord word 4583   `'ccnv 4880   dom cdm 4881    o. ccom 4885    Isom wiso 5458  OrdIsocoi 7481
This theorem is referenced by:  hartogslem1  7514  cantnfp1lem3  7639  oemapwe  7653  cantnffval2  7654  om2uzoi  11300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-oi 7482
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