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Theorem isoeq2 5493
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq2  |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  T ,  S  ( A ,  B ) ) )

Proof of Theorem isoeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 3807 . . . . 5  |-  ( R  =  T  ->  (
x R y  <->  x T
y ) )
21bibi1d 231 . . . 4  |-  ( R  =  T  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
322ralbidv 2395 . . 3  |-  ( R  =  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 T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
43anbi2d 452 . 2  |-  ( R  =  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 T y  <->  ( H `  x ) S ( H `  y ) ) ) ) )
5 df-isom 4961 . 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 4961 . 2  |-  ( H 
Isom  T ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x T y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
74, 5, 63bitr4g 221 1  |-  ( R  =  T  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  T ,  S  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   A.wral 2353   class class class wbr 3805   -1-1-onto->wf1o 4951   ` cfv 4952    Isom wiso 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-ral 2358  df-br 3806  df-isom 4961
This theorem is referenced by: (None)
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