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.

Step	Hyp	Ref	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 }
4	2, 3	nffn 5334	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2521	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1576	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2337	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2329	1 |- F/_ x X_ x e. A B

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)