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Theorem cbvixp 6616
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, A, y
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 2295 . . . . 5  |-  F/ y ( f `  x
)  e.  B
3 cbvixp.2 . . . . . 6  |-  F/_ x C
43nfel2 2295 . . . . 5  |-  F/ x
( f `  y
)  e.  C
5 fveq2 5428 . . . . . 6  |-  ( x  =  y  ->  (
f `  x )  =  ( f `  y ) )
6 cbvixp.3 . . . . . 6  |-  ( x  =  y  ->  B  =  C )
75, 6eleq12d 2211 . . . . 5  |-  ( x  =  y  ->  (
( f `  x
)  e.  B  <->  ( f `  y )  e.  C
) )
82, 4, 7cbvral 2653 . . . 4  |-  ( A. x  e.  A  (
f `  x )  e.  B  <->  A. y  e.  A  ( f `  y
)  e.  C )
98anbi2i 453 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )  <->  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C
) )
109abbii 2256 . 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 6601 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
12 dfixp 6601 . 2  |-  X_ y  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C ) }
1310, 11, 123eqtr4i 2171 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   {cab 2126   F/_wnfc 2269   A.wral 2417    Fn wfn 5125   ` cfv 5130   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fn 5133  df-fv 5138  df-ixp 6600
This theorem is referenced by:  cbvixpv  6617  mptelixpg  6635
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