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

Theorem isorel 6038
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 5455 . . 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 451 . 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 4207 . . . 4  |-  ( x  =  C  ->  (
x R y  <->  C R
y ) )
4 fveq2 5720 . . . . 5  |-  ( x  =  C  ->  ( H `  x )  =  ( H `  C ) )
54breq1d 4214 . . . 4  |-  ( x  =  C  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  C ) S ( H `  y ) ) )
63, 5bibi12d 313 . . 3  |-  ( x  =  C  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( C R y  <-> 
( H `  C
) S ( H `
 y ) ) ) )
7 breq2 4208 . . . 4  |-  ( y  =  D  ->  ( C R y  <->  C R D ) )
8 fveq2 5720 . . . . 5  |-  ( y  =  D  ->  ( H `  y )  =  ( H `  D ) )
98breq2d 4216 . . . 4  |-  ( y  =  D  ->  (
( H `  C
) S ( H `
 y )  <->  ( H `  C ) S ( H `  D ) ) )
107, 9bibi12d 313 . . 3  |-  ( y  =  D  ->  (
( C R y  <-> 
( H `  C
) S ( H `
 y ) )  <-> 
( C R D  <-> 
( H `  C
) S ( H `
 D ) ) ) )
116, 10rspc2v 3050 . 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 456 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447
This theorem is referenced by:  soisores  6039  isomin  6049  isoini  6050  isopolem  6057  isosolem  6059  weniso  6067  smoiso  6616  supisolem  7467  ordiso2  7476  cantnflt  7619  cantnfp1lem3  7628  cantnflem1b  7634  cantnflem1  7637  wemapwe  7646  cnfcomlem  7648  cnfcom  7649  cnfcom3lem  7652  fpwwe2lem6  8502  fpwwe2lem7  8503  fpwwe2lem9  8505  leisorel  11701  seqcoll  11704  seqcoll2  11705  isercoll  12453  ordthmeolem  17825  iccpnfhmeo  18962  xrhmeo  18963  dvcnvrelem1  19893  dvcvx  19896  isoun  24081  erdszelem8  24876  erdsze2lem2  24882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-isom 5455
  Copyright terms: Public domain W3C validator