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Theorem nfixp1 6542
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 6523 . 2  |-  X_ x  e.  A  B  =  { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
2 nfcv 2240 . . . . 5  |-  F/_ x
y
3 nfab1 2242 . . . . 5  |-  F/_ x { x  |  x  e.  A }
42, 3nffn 5155 . . . 4  |-  F/ x  y  Fn  { x  |  x  e.  A }
5 nfra1 2425 . . . 4  |-  F/ x A. x  e.  A  ( y `  x
)  e.  B
64, 5nfan 1512 . . 3  |-  F/ x
( y  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
y `  x )  e.  B )
76nfab 2245 . 2  |-  F/_ x { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
81, 7nfcxfr 2237 1  |-  F/_ x X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1448   {cab 2086   F/_wnfc 2227   A.wral 2375    Fn wfn 5054   ` cfv 5059   X_cixp 6522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-fun 5061  df-fn 5062  df-ixp 6523
This theorem is referenced by: (None)
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