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Theorem cbvixp 7071
Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)
Hypotheses
Ref Expression
cbvixp.1  |-  F/_ y B
cbvixp.2  |-  F/_ x C
cbvixp.3  |-  ( x  =  y  ->  B  =  C )
Assertion
Ref Expression
cbvixp  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Distinct variable group:    x, y, A
Allowed substitution hints:    B( x, y)    C( x, y)

Proof of Theorem cbvixp
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 cbvixp.1 . . . . . 6  |-  F/_ y B
21nfel2 2583 . . . . 5  |-  F/ y ( f `  x
)  e.  B
3 cbvixp.2 . . . . . 6  |-  F/_ x C
43nfel2 2583 . . . . 5  |-  F/ x
( f `  y
)  e.  C
5 fveq2 5720 . . . . . 6  |-  ( x  =  y  ->  (
f `  x )  =  ( f `  y ) )
6 cbvixp.3 . . . . . 6  |-  ( x  =  y  ->  B  =  C )
75, 6eleq12d 2503 . . . . 5  |-  ( x  =  y  ->  (
( f `  x
)  e.  B  <->  ( f `  y )  e.  C
) )
82, 4, 7cbvral 2920 . . . 4  |-  ( A. x  e.  A  (
f `  x )  e.  B  <->  A. y  e.  A  ( f `  y
)  e.  C )
98anbi2i 676 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )  <->  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C
) )
109abbii 2547 . 2  |-  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
) }  =  {
f  |  ( f  Fn  A  /\  A. y  e.  A  (
f `  y )  e.  C ) }
11 dfixp 7057 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
12 dfixp 7057 . 2  |-  X_ y  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C ) }
1310, 11, 123eqtr4i 2465 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558   A.wral 2697    Fn wfn 5441   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  cbvixpv  7072  mptelixpg  7091  ixpiunwdom  7551  prdsbas3  13695  elptr2  17598  ptunimpt  17619  ptcldmpt  17638
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-3an 938  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-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fn 5449  df-fv 5454  df-ixp 7056
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