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Theorem isoeq5 5699
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq5  |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  C ) ) )

Proof of Theorem isoeq5
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq3 5353 . . 3  |-  ( B  =  C  ->  ( H : A -1-1-onto-> B  <->  H : A -1-1-onto-> C ) )
21anbi1d 460 . 2  |-  ( B  =  C  ->  (
( 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-> C  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) ) ) ) )
3 df-isom 5127 . 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 ) ) ) )
4 df-isom 5127 . 2  |-  ( H 
Isom  R ,  S  ( A ,  C )  <-> 
( H : A -1-1-onto-> C  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
52, 3, 43bitr4g 222 1  |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   A.wral 2414   class class class wbr 3924   -1-1-onto->wf1o 5117   ` cfv 5118    Isom wiso 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-isom 5127
This theorem is referenced by:  isores3  5709  ordiso  6914  zfz1isolem1  10576  zfz1iso  10577
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