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Theorem isoeq4 6211
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq4  |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( C ,  B ) ) )

Proof of Theorem isoeq4
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq2 5804 . . 3  |-  ( A  =  C  ->  ( H : A -1-1-onto-> B  <->  H : C -1-1-onto-> B ) )
2 raleq 2986 . . . 4  |-  ( A  =  C  ->  ( A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <->  A. y  e.  C  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
32raleqbi1dv 2994 . . 3  |-  ( A  =  C  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <->  A. x  e.  C  A. y  e.  C  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
41, 3anbi12d 716 . 2  |-  ( A  =  C  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( H : C
-1-1-onto-> B  /\  A. x  e.  C  A. y  e.  C  ( x R y  <->  ( H `  x ) S ( H `  y ) ) ) ) )
5 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 ) ) ) )
6 df-isom 5590 . 2  |-  ( H 
Isom  R ,  S  ( C ,  B )  <-> 
( H : C -1-1-onto-> B  /\  A. x  e.  C  A. y  e.  C  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
74, 5, 63bitr4g 292 1  |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( C ,  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-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-isom 5590
This theorem is referenced by:  oieu  8051  oiid  8053  finnisoeu  8541  iunfictbso  8542  fz1isolem  12621  isercolllem3  13723  summolem2a  13774  prodmolem2a  13981  erdszelem1  29907  erdsze  29918  erdsze2lem1  29919  erdsze2lem2  29920  fzisoeu  37512  fourierdlem36  38000  fourierdlem112  38076  fourierdlem113  38077
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