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Theorem cbvixp 6769
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 2349 . . . . 5  |-  F/ y ( f `  x
)  e.  B
3 cbvixp.2 . . . . . 6  |-  F/_ x C
43nfel2 2349 . . . . 5  |-  F/ x
( f `  y
)  e.  C
5 fveq2 5554 . . . . . 6  |-  ( x  =  y  ->  (
f `  x )  =  ( f `  y ) )
6 cbvixp.3 . . . . . 6  |-  ( x  =  y  ->  B  =  C )
75, 6eleq12d 2264 . . . . 5  |-  ( x  =  y  ->  (
( f `  x
)  e.  B  <->  ( f `  y )  e.  C
) )
82, 4, 7cbvral 2722 . . . 4  |-  ( A. x  e.  A  (
f `  x )  e.  B  <->  A. y  e.  A  ( f `  y
)  e.  C )
98anbi2i 457 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )  <->  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C
) )
109abbii 2309 . 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 6754 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
12 dfixp 6754 . 2  |-  X_ y  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C ) }
1310, 11, 123eqtr4i 2224 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   {cab 2179   F/_wnfc 2323   A.wral 2472    Fn wfn 5249   ` cfv 5254   X_cixp 6752
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fn 5257  df-fv 5262  df-ixp 6753
This theorem is referenced by:  cbvixpv  6770  mptelixpg  6788
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