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Theorem wemoiso 6107
Description: Thus, there is at most one isomorphism between any two well-ordered sets. TODO: Shorten finnisoeu 8025. (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 445 . . . . . 6  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  R  We  A )
2 vex 2965 . . . . . . . . 9  |-  f  e. 
_V
3 isof1o 6074 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
4 f1of 5703 . . . . . . . . . 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 5655 . . . . . . . . 9  |-  ( ( f  e.  _V  /\  f : A --> B )  ->  A  e.  _V )
72, 5, 6sylancr 646 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  A  e.  _V )
87ad2antrl 710 . . . . . . 7  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  A  e.  _V )
9 exse 4575 . . . . . . 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 520 . . . . 5  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( R  We  A  /\  R Se  A ) )
12 weisoeq 6105 . . . . 5  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1311, 12sylancom 650 . . . 4  |-  ( ( R  We  A  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1413ex 425 . . 3  |-  ( R  We  A  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1514alrimivv 1643 . 2  |-  ( R  We  A  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
16 isoeq1 6068 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1716mo4 2320 . 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 205 1  |-  ( R  We  A  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    e. wcel 1727   E*wmo 2288   _Vcvv 2962   Se wse 4568    We wwe 4569   -->wf 5479   -1-1-onto->wf1o 5482    Isom wiso 5484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492
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