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Theorem isoeq3 6012
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq3  |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  T  ( A ,  B ) ) )

Proof of Theorem isoeq3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4294 . . . . 5  |-  ( S  =  T  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  x ) T ( H `  y ) ) )
21bibi2d 318 . . . 4  |-  ( S  =  T  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x R y  <-> 
( H `  x
) T ( H `
 y ) ) ) )
322ralbidv 2757 . . 3  |-  ( S  =  T  ->  ( 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  <-> 
( H `  x
) T ( H `
 y ) ) ) )
43anbi2d 703 . 2  |-  ( S  =  T  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( H : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) T ( H `  y ) ) ) ) )
5 df-isom 5427 . 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 ) ) ) )
6 df-isom 5427 . 2  |-  ( H 
Isom  R ,  T  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) T ( H `
 y ) ) ) )
74, 5, 63bitr4g 288 1  |-  ( S  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  T  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   A.wral 2715   class class class wbr 4292   -1-1-onto->wf1o 5417   ` cfv 5418    Isom wiso 5419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-cleq 2436  df-clel 2439  df-ral 2720  df-br 4293  df-isom 5427
This theorem is referenced by:  fnwelem  6687  hartogslem1  7756  leiso  12212  gtiso  25996
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