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Theorem nfixp1 6852
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 6834 . 2  |-  X_ x  e.  A  B  =  { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
2 nfcv 2432 . . . . 5  |-  F/_ x
y
3 nfab1 2434 . . . . 5  |-  F/_ x { x  |  x  e.  A }
42, 3nffn 5356 . . . 4  |-  F/ x  y  Fn  { x  |  x  e.  A }
5 nfra1 2606 . . . 4  |-  F/ x A. x  e.  A  ( y `  x
)  e.  B
64, 5nfan 1783 . . 3  |-  F/ x
( y  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
y `  x )  e.  B )
76nfab 2436 . 2  |-  F/_ x { y  |  ( y  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( y `  x )  e.  B
) }
81, 7nfcxfr 2429 1  |-  F/_ x X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    /\ wa 358    e. wcel 1696   {cab 2282   F/_wnfc 2419   A.wral 2556    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  ixpiunwdom  7321  ptbasfi  17292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274  df-ixp 6834
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