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.

Step	Hyp	Ref	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 }
4	2, 3	nffn 5155	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2425	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1512	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2245	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2237	1 |- F/_ x X_ x e. A B

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)