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Theorem ixpeq1 6702
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 5300 . . . 4  |-  ( A  =  B  ->  (
f  Fn  A  <->  f  Fn  B ) )
2 raleq 2672 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  A  ( f `  x
)  e.  C  <->  A. x  e.  B  ( f `  x )  e.  C
) )
31, 2anbi12d 473 . . 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 2295 . 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 6693 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C ) }
6 dfixp 6693 . 2  |-  X_ x  e.  B  C  =  { f  |  ( f  Fn  B  /\  A. x  e.  B  ( f `  x )  e.  C ) }
74, 5, 63eqtr4g 2235 1  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455    Fn wfn 5206   ` cfv 5211   X_cixp 6691
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-fn 5214  df-ixp 6692
This theorem is referenced by:  ixpeq1d  6703
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