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Theorem ixpeq1 7065
Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)
Assertion
Ref Expression
ixpeq1  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem ixpeq1
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fneq2 5527 . . . 4  |-  ( A  =  B  ->  (
f  Fn  A  <->  f  Fn  B ) )
2 raleq 2896 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  A  ( f `  x
)  e.  C  <->  A. x  e.  B  ( f `  x )  e.  C
) )
31, 2anbi12d 692 . . 3  |-  ( A  =  B  ->  (
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C )  <-> 
( f  Fn  B  /\  A. x  e.  B  ( f `  x
)  e.  C ) ) )
43abbidv 2549 . 2  |-  ( A  =  B  ->  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C
) }  =  {
f  |  ( f  Fn  B  /\  A. x  e.  B  (
f `  x )  e.  C ) } )
5 dfixp 7057 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C ) }
6 dfixp 7057 . 2  |-  X_ x  e.  B  C  =  { f  |  ( f  Fn  B  /\  A. x  e.  B  ( f `  x )  e.  C ) }
74, 5, 63eqtr4g 2492 1  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697    Fn wfn 5441   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  ixpeq1d  7066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-fn 5449  df-ixp 7056
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