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Theorem f1owe 5850
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 5525 . . . . . 6  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
21breq1d 4033 . . . . 5  |-  ( x  =  z  ->  (
( F `  x
) S ( F `
 y )  <->  ( F `  z ) S ( F `  y ) ) )
3 fveq2 5525 . . . . . 6  |-  ( y  =  w  ->  ( F `  y )  =  ( F `  w ) )
43breq2d 4035 . . . . 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 4284 . . . 4  |-  ( ( z  e.  A  /\  w  e.  A )  ->  ( z R w  <-> 
( F `  z
) S ( F `
 w ) ) )
76rgen2a 2609 . . 3  |-  A. z  e.  A  A. w  e.  A  ( z R w  <->  ( F `  z ) S ( F `  w ) )
8 df-isom 5264 . . . 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 5846 . . . 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 1623   A.wral 2543   class class class wbr 4023   {copab 4076    We wwe 4351   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256
This theorem is referenced by:  wemapwe  7400  dfac8b  7658  ac10ct  7661  dnwech  27145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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