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Theorem ixpeq1 7010
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 5476 . . . 4  |-  ( A  =  B  ->  (
f  Fn  A  <->  f  Fn  B ) )
2 raleq 2848 . . . 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 2502 . 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 7002 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C ) }
6 dfixp 7002 . 2  |-  X_ x  e.  B  C  =  { f  |  ( f  Fn  B  /\  A. x  e.  B  ( f `  x )  e.  C ) }
74, 5, 63eqtr4g 2445 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 1649    e. wcel 1717   {cab 2374   A.wral 2650    Fn wfn 5390   ` cfv 5395   X_cixp 7000
This theorem is referenced by:  ixpeq1d  7011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-fn 5398  df-ixp 7001
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