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Theorem wemoiso 6784
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8540. (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 458 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R  We  A )
2 vex 3081 . . . . . . . . 9  |-  f  e. 
_V
3 isof1o 6223 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
4 f1of 5823 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
53, 4syl 17 . . . . . . . . 9  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A --> B )
6 dmfex 6757 . . . . . . . . 9  |-  ( ( f  e.  _V  /\  f : A --> B )  ->  A  e.  _V )
72, 5, 6sylancr 667 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  A  e.  _V )
87ad2antrl 732 . . . . . . 7  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  A  e.  _V )
9 exse 4810 . . . . . . 7  |-  ( A  e.  _V  ->  R Se  A )
108, 9syl 17 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R Se  A )
111, 10jca 534 . . . . 5  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( R  We  A  /\  R Se  A ) )
12 weisoeq 6253 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1311, 12sylancom 671 . . . 4  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1413ex 435 . . 3  |-  ( R  We  A  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1514alrimivv 1764 . 2  |-  ( R  We  A  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
16 isoeq1 6217 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1716mo4 2311 . 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 215 1  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435    e. wcel 1867   E*wmo 2264   _Vcvv 3078   Se wse 4803    We wwe 4804   -->wf 5589   -1-1-onto->wf1o 5592    Isom wiso 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-se 4806  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602
This theorem is referenced by:  fzisoeu  37342
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