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Theorem nfixp1 6748
Description: The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
nfixp1  |-  F/_ x X_ x  e.  A  B

Proof of Theorem nfixp1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-ixp 6729 . 2  |-  X_ x  e.  A  B  =  { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
2 nfcv 2332 . . . . 5  |-  F/_ x
y
3 nfab1 2334 . . . . 5  |-  F/_ x { x  |  x  e.  A }
42, 3nffn 5334 . . . 4  |-  F/ x  y  Fn  { x  |  x  e.  A }
5 nfra1 2521 . . . 4  |-  F/ x A. x  e.  A  ( y `  x
)  e.  B
64, 5nfan 1576 . . 3  |-  F/ x
( y  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
y `  x )  e.  B )
76nfab 2337 . 2  |-  F/_ x { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
81, 7nfcxfr 2329 1  |-  F/_ x X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2160   {cab 2175   F/_wnfc 2319   A.wral 2468    Fn wfn 5233   ` cfv 5238   X_cixp 6728
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 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-fun 5240  df-fn 5241  df-ixp 6729
This theorem is referenced by: (None)
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