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Theorem isoeq4 5705
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 5357 . . 3  |-  ( A  =  C  ->  ( H : A -1-1-onto-> B  <->  H : C -1-1-onto-> B ) )
2 raleq 2626 . . . 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 2634 . . 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 464 . 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 5132 . 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 5132 . 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 222 1  |-  ( A  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( C ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   A.wral 2416   class class class wbr 3929   -1-1-onto->wf1o 5122   ` cfv 5123    Isom wiso 5124
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-isom 5132
This theorem is referenced by:  zfz1isolem1  10590  zfz1iso  10591  summodclem2a  11157  prodmodclem2a  11352
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