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Theorem cbvixp 6709
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 2332 . . . . 5  |-  F/ y ( f `  x
)  e.  B
3 cbvixp.2 . . . . . 6  |-  F/_ x C
43nfel2 2332 . . . . 5  |-  F/ x
( f `  y
)  e.  C
5 fveq2 5511 . . . . . 6  |-  ( x  =  y  ->  (
f `  x )  =  ( f `  y ) )
6 cbvixp.3 . . . . . 6  |-  ( x  =  y  ->  B  =  C )
75, 6eleq12d 2248 . . . . 5  |-  ( x  =  y  ->  (
( f `  x
)  e.  B  <->  ( f `  y )  e.  C
) )
82, 4, 7cbvral 2699 . . . 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 2293 . 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 6694 . 2  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
12 dfixp 6694 . 2  |-  X_ y  e.  A  C  =  { f  |  ( f  Fn  A  /\  A. y  e.  A  ( f `  y )  e.  C ) }
1310, 11, 123eqtr4i 2208 1  |-  X_ x  e.  A  B  =  X_ y  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   F/_wnfc 2306   A.wral 2455    Fn wfn 5207   ` cfv 5212   X_cixp 6692
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-3an 980  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-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fn 5215  df-fv 5220  df-ixp 6693
This theorem is referenced by:  cbvixpv  6710  mptelixpg  6728
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