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Theorem wemoiso 6778
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8544. (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 459 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R  We  A )
2 vex 3048 . . . . . . . . 9  |-  f  e. 
_V
3 isof1o 6216 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
4 f1of 5814 . . . . . . . . . 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 6751 . . . . . . . . 9  |-  ( ( f  e.  _V  /\  f : A --> B )  ->  A  e.  _V )
72, 5, 6sylancr 669 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  A  e.  _V )
87ad2antrl 734 . . . . . . 7  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  A  e.  _V )
9 exse 4798 . . . . . . 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 535 . . . . 5  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( R  We  A  /\  R Se  A ) )
12 weisoeq 6246 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1311, 12sylancom 673 . . . 4  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1413ex 436 . . 3  |-  ( R  We  A  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1514alrimivv 1774 . 2  |-  ( R  We  A  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
16 isoeq1 6210 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1716mo4 2346 . 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 216 1  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442    e. wcel 1887   E*wmo 2300   _Vcvv 3045   Se wse 4791    We wwe 4792   -->wf 5578   -1-1-onto->wf1o 5581    Isom wiso 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591
This theorem is referenced by:  fzisoeu  37518
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