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Theorem oieu 7254
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 733 . . . . . 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 7247 . . . . . . . 8  |-  ( ( R  We  A  /\  R Se  A )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
43adantr 451 . . . . . . 7  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( dom  F ,  A
) )
5 isocnv 5827 . . . . . . 7  |-  ( F 
Isom  _E  ,  R  ( dom  F ,  A
)  ->  `' F  Isom  R ,  _E  ( A ,  dom  F ) )
64, 5syl 15 . . . . . 6  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  `' F  Isom  R ,  _E  ( A ,  dom  F
) )
7 isotr 5833 . . . . . 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 642 . . . . 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 732 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  B )
102oicl 7244 . . . . . 6  |-  Ord  dom  F
1110a1i 10 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  Ord  dom 
F )
12 ordiso2 7230 . . . . 5  |-  ( ( ( `' F  o.  G )  Isom  _E  ,  _E  ( B ,  dom  F )  /\  Ord  B  /\  Ord  dom  F )  ->  B  =  dom  F
)
138, 9, 11, 12syl3anc 1182 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  B  =  dom  F )
14 ordwe 4405 . . . . . 6  |-  ( Ord 
B  ->  _E  We  B )
1514ad2antrl 708 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E  We  B )
16 epse 4376 . . . . . 6  |-  _E Se  B
1716a1i 10 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  _E Se  B )
18 isoeq4 5819 . . . . . . 7  |-  ( B  =  dom  F  -> 
( F  Isom  _E  ,  R  ( B ,  A )  <->  F  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
1913, 18syl 15 . . . . . 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 223 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  F  Isom  _E  ,  R  ( B ,  A ) )
21 weisoeq 5853 . . . . 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 1183 . . . 4  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  G  =  F )
2313, 22jca 518 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) )  ->  ( B  =  dom  F  /\  G  =  F )
)
2423ex 423 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) )  ->  ( B  =  dom  F  /\  G  =  F ) ) )
253, 10jctil 523 . . 3  |-  ( ( R  We  A  /\  R Se  A )  ->  ( Ord  dom  F  /\  F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
26 ordeq 4399 . . . . 5  |-  ( B  =  dom  F  -> 
( Ord  B  <->  Ord  dom  F
) )
2726adantr 451 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( Ord  B  <->  Ord  dom  F ) )
28 isoeq4 5819 . . . . 5  |-  ( B  =  dom  F  -> 
( G  Isom  _E  ,  R  ( B ,  A )  <->  G  Isom  _E  ,  R  ( dom 
F ,  A ) ) )
29 isoeq1 5816 . . . . 5  |-  ( G  =  F  ->  ( G  Isom  _E  ,  R  ( dom  F ,  A
)  <->  F  Isom  _E  ,  R  ( dom  F ,  A ) ) )
3028, 29sylan9bb 680 . . . 4  |-  ( ( B  =  dom  F  /\  G  =  F
)  ->  ( G  Isom  _E  ,  R  ( B ,  A )  <-> 
F  Isom  _E  ,  R  ( dom  F ,  A
) ) )
3127, 30anbi12d 691 . . 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 213 . 2  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( B  =  dom  F  /\  G  =  F )  ->  ( Ord  B  /\  G  Isom  _E  ,  R  ( B ,  A ) ) ) )
3324, 32impbid 183 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 176    /\ wa 358    = wceq 1623    _E cep 4303   Se wse 4350    We wwe 4351   Ord word 4391   `'ccnv 4688   dom cdm 4689    o. ccom 4693    Isom wiso 5256  OrdIsocoi 7224
This theorem is referenced by:  hartogslem1  7257  cantnfp1lem3  7382  oemapwe  7396  cantnffval2  7397  om2uzoi  11018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-oi 7225
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