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Theorem isorel 5479
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 4941 . . 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 269 . 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 3796 . . . 4  |-  ( x  =  C  ->  (
x R y  <->  C R
y ) )
4 fveq2 5209 . . . . 5  |-  ( x  =  C  ->  ( H `  x )  =  ( H `  C ) )
54breq1d 3803 . . . 4  |-  ( x  =  C  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  C ) S ( H `  y ) ) )
63, 5bibi12d 233 . . 3  |-  ( x  =  C  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( C R y  <-> 
( H `  C
) S ( H `
 y ) ) ) )
7 breq2 3797 . . . 4  |-  ( y  =  D  ->  ( C R y  <->  C R D ) )
8 fveq2 5209 . . . . 5  |-  ( y  =  D  ->  ( H `  y )  =  ( H `  D ) )
98breq2d 3805 . . . 4  |-  ( y  =  D  ->  (
( H `  C
) S ( H `
 y )  <->  ( H `  C ) S ( H `  D ) ) )
107, 9bibi12d 233 . . 3  |-  ( y  =  D  ->  (
( C R y  <-> 
( H `  C
) S ( H `
 y ) )  <-> 
( C R D  <-> 
( H `  C
) S ( H `
 D ) ) ) )
116, 10rspc2v 2714 . 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 275 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   class class class wbr 3793   -1-1-onto->wf1o 4931   ` cfv 4932    Isom wiso 4933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-isom 4941
This theorem is referenced by:  isoresbr  5480  isoini  5488  isopolem  5492  isosolem  5494  smoiso  5951  isotilem  6478  supisolem  6480  ordiso2  6505
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