Theorem nfixp1 6955

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 6936	. 2 |- X_ x e. A B = { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
2		nfcv 2386	. . . . 5 |- F/_ x y
3		nfab1 2388	. . . . 5 |- F/_ x { x | x e. A }
4	2, 3	nffn 5454	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2575	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1614	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2391	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2383	1 |- F/_ x X_ x e. A B

Syntax hints:   /\ wa 104   e. wcel 2205   {cab 2220   F/_wnfc 2373   A.wral 2522   Fn wfn 5349   ` cfv 5354   X_cixp 6935

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-fun 5356  df-fn 5357  df-ixp 6936

This theorem is referenced by: (None)