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Theorem ixpeq1 6827
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 5334 . . . 4  |-  ( A  =  B  ->  (
f  Fn  A  <->  f  Fn  B ) )
2 raleq 2736 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  A  ( f `  x
)  e.  C  <->  A. x  e.  B  ( f `  x )  e.  C
) )
31, 2anbi12d 691 . . 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 2397 . 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 6819 . 2  |-  X_ x  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C ) }
6 dfixp 6819 . 2  |-  X_ x  e.  B  C  =  { f  |  ( f  Fn  B  /\  A. x  e.  B  ( f `  x )  e.  C ) }
74, 5, 63eqtr4g 2340 1  |-  ( A  =  B  ->  X_ x  e.  A  C  =  X_ x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  ixpeq1d  6828  elixpsn  6855  ixpsnf1o  6856  dfac9  7762  dprdval  15238  ptval  17265  dfac14  17312  ptuncnv  17498  ptunhmeo  17499  prjmapcp2  25170  istopx  25547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-fn 5258  df-ixp 6818
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