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Theorem oieu 7997
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 758 . . . . . 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 7990 . . . . . . . 8  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
43adantr 463 . . . . . . 7  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
5 isocnv 6208 . . . . . . 7  |-  ( F 
Isom  _E  ,  R  ( dom  F ,  A
)  ->  `' F  Isom  R ,  _E  ( A ,  dom  F ) )
64, 5syl 17 . . . . . 6  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  `' F  Isom  R ,  _E  ( A ,  dom  F
) )
7 isotr 6214 . . . . . 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 659 . . . . 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 756 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  B )
102oicl 7987 . . . . . 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 7973 . . . . 5  |-  ( ( ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F )  /\  Ord  B  /\  Ord  dom  F )  ->  B  =  dom  F
)
138, 9, 11, 12syl3anc 1230 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  B  =  dom  F )
14 ordwe 5422 . . . . . 6  |-  ( Ord 
B  ->  _E  We  B )
1514ad2antrl 726 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E  We  B )
16 epse 4805 . . . . . 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 6200 . . . . . . 7  |-  ( B  =  dom  F  -> 
( F  Isom  _E  ,  R  ( B ,  A )  <->  F  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
1913, 18syl 17 . . . . . 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 6233 . . . . 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 1231 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  =  F )
2313, 22jca 530 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( B  =  dom  F  /\  G  =  F )
)
2423ex 432 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  ->  ( B  =  dom  F  /\  G  =  F ) ) )
253, 10jctil 535 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  /\  F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
26 ordeq 5416 . . . . 5  |-  ( B  =  dom  F  -> 
( Ord  B  <->  Ord  dom  F
) )
2726adantr 463 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( Ord  B  <->  Ord  dom  F ) )
28 isoeq4 6200 . . . . 5  |-  ( B  =  dom  F  -> 
( G  Isom  _E  ,  R  ( B ,  A )  <->  G  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
29 isoeq1 6197 . . . . 5  |-  ( G  =  F  ->  ( G  Isom  _E  ,  R  ( dom  F ,  A
)  <->  F  Isom  _E  ,  R  ( dom  F ,  A ) ) )
3028, 29sylan9bb 698 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( G  Isom  _E  ,  R  ( B ,  A )  <-> 
F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
3127, 30anbi12d 709 . . 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 367    = wceq 1405    _E cep 4731   Se wse 4779    We wwe 4780   `'ccnv 4821   dom cdm 4822    o. ccom 4826   Ord word 5408    Isom wiso 5569  OrdIsocoi 7967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-wrecs 7012  df-recs 7074  df-oi 7968
This theorem is referenced by:  hartogslem1  8000  cantnfp1lem3  8130  oemapwe  8144  cantnffval2  8145  cantnfp1lem3OLD  8156  oemapweOLD  8166  cantnffval2OLD  8167  om2uzoi  12105
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