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Theorem wemoiso 5857
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 7742. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wemoiso  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Distinct variable groups:    R, f    A, f    S, f    B, f

Proof of Theorem wemoiso
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R  We  A )
2 vex 2793 . . . . . . . . 9  |-  f  e. 
_V
3 isof1o 5824 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
4 f1of 5474 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
53, 4syl 15 . . . . . . . . 9  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A --> B )
6 dmfex 5426 . . . . . . . . 9  |-  ( ( f  e.  _V  /\  f : A --> B )  ->  A  e.  _V )
72, 5, 6sylancr 644 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  A  e.  _V )
87ad2antrl 708 . . . . . . 7  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  A  e.  _V )
9 exse 4359 . . . . . . 7  |-  ( A  e.  _V  ->  R Se  A )
108, 9syl 15 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R Se  A )
111, 10jca 518 . . . . 5  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( R  We  A  /\  R Se  A ) )
12 weisoeq 5855 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1311, 12sylancom 648 . . . 4  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1413ex 423 . . 3  |-  ( R  We  A  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1514alrimivv 1620 . 2  |-  ( R  We  A  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
16 isoeq1 5818 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1716mo4 2178 . 2  |-  ( E* f  f  Isom  R ,  S  ( A ,  B )  <->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
1815, 17sylibr 203 1  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1529    = wceq 1625    e. wcel 1686   E*wmo 2146   _Vcvv 2790   Se wse 4352    We wwe 4353   -->wf 5253   -1-1-onto->wf1o 5256    Isom wiso 5258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266
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