Theorem nfixp1 6684

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 6665	. 2 |- X_ x e. A B = { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
2		nfcv 2308	. . . . 5 |- F/_ x y
3		nfab1 2310	. . . . 5 |- F/_ x { x | x e. A }
4	2, 3	nffn 5284	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2497	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1553	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2313	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2305	1 |- F/_ x X_ x e. A B

Syntax hints:   /\ wa 103   e. wcel 2136   {cab 2151   F/_wnfc 2295   A.wral 2444   Fn wfn 5183   ` cfv 5188   X_cixp 6664

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191  df-ixp 6665

This theorem is referenced by: (None)