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Theorem isoeq4 5829
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 5472 . . 3  |-  ( A  =  C  ->  ( H : A -1-1-onto-> B  <->  H : C -1-1-onto-> B ) )
2 raleq 2686 . . . 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 2694 . . 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 473 . 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 5247 . 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 5247 . 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 223 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 104    <-> wb 105    = wceq 1364   A.wral 2468   class class class wbr 4021   -1-1-onto->wf1o 5237   ` cfv 5238    Isom wiso 5239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-isom 5247
This theorem is referenced by:  zfz1isolem1  10861  zfz1iso  10862  summodclem2a  11430  prodmodclem2a  11625
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