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Theorem isorel 6217
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 5591 . . 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 466 . 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 4405 . . . 4  |-  ( x  =  C  ->  (
x R y  <->  C R
y ) )
4 fveq2 5865 . . . . 5  |-  ( x  =  C  ->  ( H `  x )  =  ( H `  C ) )
54breq1d 4412 . . . 4  |-  ( x  =  C  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  C ) S ( H `  y ) ) )
63, 5bibi12d 323 . . 3  |-  ( x  =  C  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( C R y  <-> 
( H `  C
) S ( H `
 y ) ) ) )
7 breq2 4406 . . . 4  |-  ( y  =  D  ->  ( C R y  <->  C R D ) )
8 fveq2 5865 . . . . 5  |-  ( y  =  D  ->  ( H `  y )  =  ( H `  D ) )
98breq2d 4414 . . . 4  |-  ( y  =  D  ->  (
( H `  C
) S ( H `
 y )  <->  ( H `  C ) S ( H `  D ) ) )
107, 9bibi12d 323 . . 3  |-  ( y  =  D  ->  (
( C R y  <-> 
( H `  C
) S ( H `
 y ) )  <-> 
( C R D  <-> 
( H `  C
) S ( H `
 D ) ) ) )
116, 10rspc2v 3159 . 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 472 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   class class class wbr 4402   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-isom 5591
This theorem is referenced by:  soisores  6218  isomin  6228  isoini  6229  isopolem  6236  isosolem  6238  weniso  6245  smoiso  7081  supisolem  7989  ordiso2  8030  cantnflt  8177  cantnfp1lem3  8185  cantnflem1b  8191  cantnflem1  8194  wemapwe  8202  cnfcomlem  8204  cnfcom  8205  cnfcom3lem  8208  fpwwe2lem6  9060  fpwwe2lem7  9061  fpwwe2lem9  9063  leisorel  12623  seqcoll  12627  seqcoll2  12628  isercoll  13731  ordthmeolem  20816  iccpnfhmeo  21973  xrhmeo  21974  dvcnvrelem1  22969  dvcvx  22972  isoun  28282  erdszelem8  29921  erdsze2lem2  29927  fourierdlem20  37989  fourierdlem46  38016  fourierdlem50  38020  fourierdlem63  38033  fourierdlem64  38034  fourierdlem65  38035  fourierdlem76  38046  fourierdlem79  38049  fourierdlem102  38072  fourierdlem103  38073  fourierdlem104  38074  fourierdlem114  38084
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