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Theorem oieu 7758
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 7751 . . . . . . . 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 6026 . . . . . . 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 6032 . . . . . 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 7748 . . . . . 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 7734 . . . . 5  |-  ( ( ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F )  /\  Ord  B  /\  Ord  dom  F )  ->  B  =  dom  F
)
138, 9, 11, 12syl3anc 1218 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  B  =  dom  F )
14 ordwe 4737 . . . . . 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 4708 . . . . . 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 6018 . . . . . . 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 6051 . . . . 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 1219 . . . 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 4731 . . . . 5  |-  ( B  =  dom  F  -> 
( Ord  B  <->  Ord  dom  F
) )
2726adantr 465 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( Ord  B  <->  Ord  dom  F ) )
28 isoeq4 6018 . . . . 5  |-  ( B  =  dom  F  -> 
( G  Isom  _E  ,  R  ( B ,  A )  <->  G  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
29 isoeq1 6015 . . . . 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 1369    _E cep 4635   Se wse 4682    We wwe 4683   Ord word 4723   `'ccnv 4844   dom cdm 4845    o. ccom 4849    Isom wiso 5424  OrdIsocoi 7728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-recs 6837  df-oi 7729
This theorem is referenced by:  hartogslem1  7761  cantnfp1lem3  7893  oemapwe  7907  cantnffval2  7908  cantnfp1lem3OLD  7919  oemapweOLD  7929  cantnffval2OLD  7930  om2uzoi  11783
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