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Theorem nfixp1 7012
Description: The index variable in an indexed cross 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 6994 . 2  |-  X_ x  e.  A  B  =  { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
2 nfcv 2517 . . . . 5  |-  F/_ x
y
3 nfab1 2519 . . . . 5  |-  F/_ x { x  |  x  e.  A }
42, 3nffn 5475 . . . 4  |-  F/ x  y  Fn  { x  |  x  e.  A }
5 nfra1 2693 . . . 4  |-  F/ x A. x  e.  A  ( y `  x
)  e.  B
64, 5nfan 1836 . . 3  |-  F/ x
( y  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
y `  x )  e.  B )
76nfab 2521 . 2  |-  F/_ x { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
81, 7nfcxfr 2514 1  |-  F/_ x X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1717   {cab 2367   F/_wnfc 2504   A.wral 2643    Fn wfn 5383   ` cfv 5388   X_cixp 6993
This theorem is referenced by:  ixpiunwdom  7486  ptbasfi  17528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ral 2648  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-br 4148  df-opab 4202  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-fun 5390  df-fn 5391  df-ixp 6994
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