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Theorem isoeq5 5836
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 5481 . . 3  |-  ( B  =  C  ->  ( H : A -1-1-onto-> B  <->  H : A -1-1-onto-> C ) )
21anbi1d 685 . 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 5280 . 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 5280 . 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 279 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    <-> wb 176    /\ wa 358    = wceq 1632   A.wral 2556   class class class wbr 4039   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  isores3  5848  ordiso  7247  ordtypelem9  7257  ordtypelem10  7258  oiid  7272  iunfictbso  7757  ltweuz  11040  fz1isolem  11415  dvgt0lem2  19366  erdszelem1  23737  erdsze  23748  erdsze2lem1  23749  erdsze2lem2  23750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-isom 5280
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