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Theorem f1owe 5866
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 5541 . . . . . 6  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21breq1d 4049 . . . . 5  |-  ( x  =  z  ->  (
( F `  x
) S ( F `
 y )  <->  ( F `  z ) S ( F `  y ) ) )
3 fveq2 5541 . . . . . 6  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
43breq2d 4051 . . . . 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 4300 . . . 4  |-  ( ( z  e.  A  /\  w  e.  A )  ->  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) )
76rgen2a 2622 . . 3  |-  A. z  e.  A  A. w  e.  A  ( z R w  <->  ( F `  z ) S ( F `  w ) )
8 df-isom 5280 . . . 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 5862 . . . 4  |-  ( F 
Isom  R ,  S  ( A ,  B )  ->  ( R  We  A 
<->  S  We  B ) )
108, 9sylbir 204 . . 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 652 . 2  |-  ( F : A -1-1-onto-> B  ->  ( R  We  A  <->  S  We  B
) )
1211biimprd 214 1  |-  ( F : A -1-1-onto-> B  ->  ( S  We  B  ->  R  We  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632   A.wral 2556   class class class wbr 4039   {copab 4092    We wwe 4367   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  wemapwe  7416  dfac8b  7674  ac10ct  7677  dnwech  27248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280
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