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

Theorem f1owe 6262
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 5879 . . . . . 6  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21breq1d 4405 . . . . 5  |-  ( x  =  z  ->  (
( F `  x
) S ( F `
 y )  <->  ( F `  z ) S ( F `  y ) ) )
3 fveq2 5879 . . . . . 6  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
43breq2d 4407 . . . . 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 4720 . . . 4  |-  ( ( z  e.  A  /\  w  e.  A )  ->  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) )
76rgen2a 2820 . . 3  |-  A. z  e.  A  A. w  e.  A  ( z R w  <->  ( F `  z ) S ( F `  w ) )
8 df-isom 5598 . . . 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 6258 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )
108, 9sylbir 218 . . 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 685 . 2  |-  ( F : A -1-1-onto-> B  ->  ( R  We  A  <->  S  We  B
) )
1211biimprd 231 1  |-  ( F : A -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   A.wral 2756   class class class wbr 4395   {copab 4453    We wwe 4797   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598
This theorem is referenced by:  wemapwe  8220  dfac8b  8480  ac10ct  8483  dnwech  35977
  Copyright terms: Public domain W3C validator