MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wemoiso2 Structured version   Unicode version

Theorem wemoiso2 6558
Description: Thus, there is at most one isomorphism between any two well-ordered sets. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
wemoiso2  |-  ( S  We  B  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Distinct variable groups:    R, f    A, f    S, f    B, f

Proof of Theorem wemoiso2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . . 6  |-  ( ( S  We  B  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  S  We  B )
2 isof1o 6011 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
3 f1ofo 5643 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
4 forn 5618 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
52, 3, 43syl 20 . . . . . . . . 9  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  ran  f  =  B )
6 vex 2970 . . . . . . . . . 10  |-  f  e. 
_V
76rnex 6507 . . . . . . . . 9  |-  ran  f  e.  _V
85, 7syl6eqelr 2527 . . . . . . . 8  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  B  e.  _V )
98ad2antrl 727 . . . . . . 7  |-  ( ( S  We  B  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  B  e.  _V )
10 exse 4679 . . . . . . 7  |-  ( B  e.  _V  ->  S Se  B )
119, 10syl 16 . . . . . 6  |-  ( ( S  We  B  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  S Se  B )
121, 11jca 532 . . . . 5  |-  ( ( S  We  B  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  ( S  We  B  /\  S Se  B ) )
13 weisoeq2 6042 . . . . 5  |-  ( ( ( S  We  B  /\  S Se  B )  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1412, 13sylancom 667 . . . 4  |-  ( ( S  We  B  /\  ( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) ) )  ->  f  =  g )
1514ex 434 . . 3  |-  ( S  We  B  ->  (
( f  Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  ->  f  =  g ) )
1615alrimivv 1686 . 2  |-  ( S  We  B  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
17 isoeq1 6005 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1817mo4 2315 . 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 ) )
1916, 18sylibr 212 1  |-  ( S  We  B  ->  E* f  f  Isom  R ,  S  ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   E*wmo 2253   _Vcvv 2967   Se wse 4672    We wwe 4673   ran crn 4836   -onto->wfo 5411   -1-1-onto->wf1o 5412    Isom wiso 5414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422
This theorem is referenced by:  finnisoeu  8275
  Copyright terms: Public domain W3C validator