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Theorem isorel 5829
Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isorel  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  e.  A  /\  D  e.  A )
)  ->  ( C R D  <->  ( H `  C ) S ( H `  D ) ) )

Proof of Theorem isorel
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 5244 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
21simprbi 275 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )
3 breq1 4021 . . . 4  |-  ( x  =  C  ->  (
x R y  <->  C R
y ) )
4 fveq2 5534 . . . . 5  |-  ( x  =  C  ->  ( H `  x )  =  ( H `  C ) )
54breq1d 4028 . . . 4  |-  ( x  =  C  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  C ) S ( H `  y ) ) )
63, 5bibi12d 235 . . 3  |-  ( x  =  C  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( C R y  <-> 
( H `  C
) S ( H `
 y ) ) ) )
7 breq2 4022 . . . 4  |-  ( y  =  D  ->  ( C R y  <->  C R D ) )
8 fveq2 5534 . . . . 5  |-  ( y  =  D  ->  ( H `  y )  =  ( H `  D ) )
98breq2d 4030 . . . 4  |-  ( y  =  D  ->  (
( H `  C
) S ( H `
 y )  <->  ( H `  C ) S ( H `  D ) ) )
107, 9bibi12d 235 . . 3  |-  ( y  =  D  ->  (
( C R y  <-> 
( H `  C
) S ( H `
 y ) )  <-> 
( C R D  <-> 
( H `  C
) S ( H `
 D ) ) ) )
116, 10rspc2v 2869 . 2  |-  ( ( C  e.  A  /\  D  e.  A )  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) )  ->  ( C R D  <->  ( H `  C ) S ( H `  D ) ) ) )
122, 11mpan9 281 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  e.  A  /\  D  e.  A )
)  ->  ( C R D  <->  ( H `  C ) S ( H `  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   class class class wbr 4018   -1-1-onto->wf1o 5234   ` cfv 5235    Isom wiso 5236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-isom 5244
This theorem is referenced by:  isoresbr  5830  isoini  5839  isopolem  5843  isosolem  5845  smoiso  6326  isotilem  7034  supisolem  7036  ordiso2  7063  leisorel  10848  zfz1isolemiso  10850  seq3coll  10853
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