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Theorem isoeq1 6208
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq1  |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
G  Isom  R ,  S  ( A ,  B ) ) )

Proof of Theorem isoeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq1 5803 . . 3  |-  ( H  =  G  ->  ( H : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
2 fveq1 5862 . . . . . 6  |-  ( H  =  G  ->  ( H `  x )  =  ( G `  x ) )
3 fveq1 5862 . . . . . 6  |-  ( H  =  G  ->  ( H `  y )  =  ( G `  y ) )
42, 3breq12d 4414 . . . . 5  |-  ( H  =  G  ->  (
( H `  x
) S ( H `
 y )  <->  ( G `  x ) S ( G `  y ) ) )
54bibi2d 320 . . . 4  |-  ( H  =  G  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
652ralbidv 2831 . . 3  |-  ( H  =  G  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <->  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
71, 6anbi12d 716 . 2  |-  ( H  =  G  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( G : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( G `  x ) S ( G `  y ) ) ) ) )
8 df-isom 5590 . 2  |-  ( 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 ) ) ) )
9 df-isom 5590 . 2  |-  ( G 
Isom  R ,  S  ( A ,  B )  <-> 
( G : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
107, 8, 93bitr4g 292 1  |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
G  Isom  R ,  S  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443   A.wral 2736   class class class wbr 4401   -1-1-onto->wf1o 5580   ` cfv 5581    Isom wiso 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590
This theorem is referenced by:  isores1  6223  wemoiso  6775  wemoiso2  6776  ordiso  8028  oieu  8051  finnisoeu  8541  iunfictbso  8542  infrenegsup  10588  infmsupOLD  10589  ltweuz  12172  fz1isolem  12621  isercolllem2  13722  isercoll  13724  dvgt0lem2  22948  efcvx  23397  relogiso  23540  logccv  23601  erdszelem1  29907  erdsze  29918  erdsze2lem2  29920  fzisoeu  37512  fourierdlem36  38000  fourierdlem96  38060  fourierdlem97  38061  fourierdlem98  38062  fourierdlem99  38063  fourierdlem105  38069  fourierdlem106  38070  fourierdlem108  38072  fourierdlem110  38074  fourierdlem112  38076  fourierdlem113  38077  fourierdlem115  38079
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