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Theorem isoeq4 6112
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 5731 . . 3  |-  ( A  =  C  ->  ( H : A -1-1-onto-> B  <->  H : C -1-1-onto-> B ) )
2 raleq 3013 . . . 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 3021 . . 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 710 . 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 5525 . 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 5525 . 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 288 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 184    /\ wa 369    = wceq 1370   A.wral 2795   class class class wbr 4390   -1-1-onto->wf1o 5515   ` cfv 5516    Isom wiso 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-isom 5525
This theorem is referenced by:  oieu  7854  oiid  7856  finnisoeu  8384  iunfictbso  8385  fz1isolem  12316  isercolllem3  13246  summolem2a  13294  erdszelem1  27213  erdsze  27224  erdsze2lem1  27225  erdsze2lem2  27226  prodmolem2a  27581
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