Theorem nfixp1 6675

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 6656	. 2 |- X_ x e. A B = { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
2		nfcv 2306	. . . . 5 |- F/_ x y
3		nfab1 2308	. . . . 5 |- F/_ x { x | x e. A }
4	2, 3	nffn 5278	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2495	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1552	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2311	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2303	1 |- F/_ x X_ x e. A B

Syntax hints:   /\ wa 103   e. wcel 2135   {cab 2150   F/_wnfc 2293   A.wral 2442   Fn wfn 5177   ` cfv 5182   X_cixp 6655

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-fun 5184  df-fn 5185  df-ixp 6656

This theorem is referenced by: (None)