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Theorem wemoiso 6045
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 7958. (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 444 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R  We  A )
2 vex 2927 . . . . . . . . 9  |-  f  e. 
_V
3 isof1o 6012 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
4 f1of 5641 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
53, 4syl 16 . . . . . . . . 9  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A --> B )
6 dmfex 5593 . . . . . . . . 9  |-  ( ( f  e.  _V  /\  f : A --> B )  ->  A  e.  _V )
72, 5, 6sylancr 645 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  A  e.  _V )
87ad2antrl 709 . . . . . . 7  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  A  e.  _V )
9 exse 4514 . . . . . . 7  |-  ( A  e.  _V  ->  R Se  A )
108, 9syl 16 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R Se  A )
111, 10jca 519 . . . . 5  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( R  We  A  /\  R Se  A ) )
12 weisoeq 6043 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1311, 12sylancom 649 . . . 4  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1413ex 424 . . 3  |-  ( R  We  A  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1514alrimivv 1639 . 2  |-  ( R  We  A  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
16 isoeq1 6006 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1716mo4 2295 . 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 204 1  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    e. wcel 1721   E*wmo 2263   _Vcvv 2924   Se wse 4507    We wwe 4508   -->wf 5417   -1-1-onto->wf1o 5420    Isom wiso 5422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430
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