MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isorel Structured version   Unicode version

Theorem isorel 6208
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 5595 . . 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 464 . 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 4450 . . . 4  |-  ( x  =  C  ->  (
x R y  <->  C R
y ) )
4 fveq2 5864 . . . . 5  |-  ( x  =  C  ->  ( H `  x )  =  ( H `  C ) )
54breq1d 4457 . . . 4  |-  ( x  =  C  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  C ) S ( H `  y ) ) )
63, 5bibi12d 321 . . 3  |-  ( x  =  C  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( C R y  <-> 
( H `  C
) S ( H `
 y ) ) ) )
7 breq2 4451 . . . 4  |-  ( y  =  D  ->  ( C R y  <->  C R D ) )
8 fveq2 5864 . . . . 5  |-  ( y  =  D  ->  ( H `  y )  =  ( H `  D ) )
98breq2d 4459 . . . 4  |-  ( y  =  D  ->  (
( H `  C
) S ( H `
 y )  <->  ( H `  C ) S ( H `  D ) ) )
107, 9bibi12d 321 . . 3  |-  ( y  =  D  ->  (
( C R y  <-> 
( H `  C
) S ( H `
 y ) )  <-> 
( C R D  <-> 
( H `  C
) S ( H `
 D ) ) ) )
116, 10rspc2v 3223 . 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 469 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   class class class wbr 4447   -1-1-onto->wf1o 5585   ` cfv 5586    Isom wiso 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-isom 5595
This theorem is referenced by:  soisores  6209  isomin  6219  isoini  6220  isopolem  6227  isosolem  6229  weniso  6236  smoiso  7030  supisolem  7927  ordiso2  7936  cantnflt  8087  cantnfp1lem3  8095  cantnflem1b  8101  cantnflem1  8104  cantnfltOLD  8117  cantnfp1lem3OLD  8121  cantnflem1bOLD  8124  cantnflem1OLD  8127  wemapwe  8135  wemapweOLD  8136  cnfcomlem  8139  cnfcom  8140  cnfcom3lem  8143  cnfcomlemOLD  8147  cnfcomOLD  8148  cnfcom3lemOLD  8151  fpwwe2lem6  9009  fpwwe2lem7  9010  fpwwe2lem9  9012  leisorel  12471  seqcoll  12474  seqcoll2  12475  isercoll  13449  ordthmeolem  20037  iccpnfhmeo  21180  xrhmeo  21181  dvcnvrelem1  22153  dvcvx  22156  isoun  27192  erdszelem8  28282  erdsze2lem2  28288  fourierdlem20  31427  fourierdlem46  31453  fourierdlem50  31457  fourierdlem63  31470  fourierdlem64  31471  fourierdlem65  31472  fourierdlem76  31483  fourierdlem79  31486  fourierdlem102  31509  fourierdlem103  31510  fourierdlem104  31511  fourierdlem114  31521
  Copyright terms: Public domain W3C validator