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Theorem nfixp1 6615
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 6596 . 2  |-  X_ x  e.  A  B  =  { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
2 nfcv 2281 . . . . 5  |-  F/_ x
y
3 nfab1 2283 . . . . 5  |-  F/_ x { x  |  x  e.  A }
42, 3nffn 5222 . . . 4  |-  F/ x  y  Fn  { x  |  x  e.  A }
5 nfra1 2466 . . . 4  |-  F/ x A. x  e.  A  ( y `  x
)  e.  B
64, 5nfan 1544 . . 3  |-  F/ x
( y  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
y `  x )  e.  B )
76nfab 2286 . 2  |-  F/_ x { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
81, 7nfcxfr 2278 1  |-  F/_ x X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1480   {cab 2125   F/_wnfc 2268   A.wral 2416    Fn wfn 5121   ` cfv 5126   X_cixp 6595
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-fun 5128  df-fn 5129  df-ixp 6596
This theorem is referenced by: (None)
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