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Theorem f1owe 6012
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 5668 . . . . . 6  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21breq1d 4163 . . . . 5  |-  ( x  =  z  ->  (
( F `  x
) S ( F `
 y )  <->  ( F `  z ) S ( F `  y ) ) )
3 fveq2 5668 . . . . . 6  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
43breq2d 4165 . . . . 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 4415 . . . 4  |-  ( ( z  e.  A  /\  w  e.  A )  ->  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) )
76rgen2a 2715 . . 3  |-  A. z  e.  A  A. w  e.  A  ( z R w  <->  ( F `  z ) S ( F `  w ) )
8 df-isom 5403 . . . 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 6008 . . . 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 1649   A.wral 2649   class class class wbr 4153   {copab 4206    We wwe 4481   -1-1-onto->wf1o 5393   ` cfv 5394    Isom wiso 5395
This theorem is referenced by:  wemapwe  7587  dfac8b  7845  ac10ct  7848  dnwech  26814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403
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