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

Theorem wemoiso2 6676
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 6128 . . . . . . . . . 10  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  f : A -1-1-onto-> B
)
3 f1ofo 5759 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
4 forn 5734 . . . . . . . . . 10  |-  ( f : A -onto-> B  ->  ran  f  =  B
)
52, 3, 43syl 20 . . . . . . . . 9  |-  ( f 
Isom  R ,  S  ( A ,  B )  ->  ran  f  =  B )
6 vex 3081 . . . . . . . . . 10  |-  f  e. 
_V
76rnex 6625 . . . . . . . . 9  |-  ran  f  e.  _V
85, 7syl6eqelr 2551 . . . . . . . 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 4795 . . . . . . 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 6159 . . . . 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 1687 . 2  |-  ( S  We  B  ->  A. f A. g ( ( f 
Isom  R ,  S  ( A ,  B )  /\  g  Isom  R ,  S  ( A ,  B ) )  -> 
f  =  g ) )
17 isoeq1 6122 . . 3  |-  ( f  =  g  ->  (
f  Isom  R ,  S  ( A ,  B )  <->  g  Isom  R ,  S  ( A ,  B ) ) )
1817mo4 2326 . 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 1368    = wceq 1370    e. wcel 1758   E*wmo 2263   _Vcvv 3078   Se wse 4788    We wwe 4789   ran crn 4952   -onto->wfo 5527   -1-1-onto->wf1o 5528    Isom wiso 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538
This theorem is referenced by:  finnisoeu  8397
  Copyright terms: Public domain W3C validator