MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1owe Structured version   Unicode version

Theorem f1owe 6065
Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
Hypothesis
Ref Expression
f1owe.1  |-  R  =  { <. x ,  y
>.  |  ( F `  x ) S ( F `  y ) }
Assertion
Ref Expression
f1owe  |-  ( F : A -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
Distinct variable groups:    x, y, S    x, F, y
Allowed substitution hints:    A( x, y)    B( x, y)    R( x, y)

Proof of Theorem f1owe
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . 6  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21breq1d 4214 . . . . 5  |-  ( x  =  z  ->  (
( F `  x
) S ( F `
 y )  <->  ( F `  z ) S ( F `  y ) ) )
3 fveq2 5720 . . . . . 6  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
43breq2d 4216 . . . . 5  |-  ( y  =  w  ->  (
( F `  z
) S ( F `
 y )  <->  ( F `  z ) S ( F `  w ) ) )
5 f1owe.1 . . . . 5  |-  R  =  { <. x ,  y
>.  |  ( F `  x ) S ( F `  y ) }
62, 4, 5brabg 4466 . . . 4  |-  ( ( z  e.  A  /\  w  e.  A )  ->  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) )
76rgen2a 2764 . . 3  |-  A. z  e.  A  A. w  e.  A  ( z R w  <->  ( F `  z ) S ( F `  w ) )
8 df-isom 5455 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  <-> 
( F : A -1-1-onto-> B  /\  A. z  e.  A  A. w  e.  A  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) ) )
9 isowe 6061 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )
108, 9sylbir 205 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  A. z  e.  A  A. w  e.  A  (
z R w  <->  ( F `  z ) S ( F `  w ) ) )  ->  ( R  We  A  <->  S  We  B ) )
117, 10mpan2 653 . 2  |-  ( F : A -1-1-onto-> B  ->  ( R  We  A  <->  S  We  B
) )
1211biimprd 215 1  |-  ( F : A -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652   A.wral 2697   class class class wbr 4204   {copab 4257    We wwe 4532   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447
This theorem is referenced by:  wemapwe  7646  dfac8b  7904  ac10ct  7907  dnwech  27114
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455
  Copyright terms: Public domain W3C validator