Theorem nfixp1 6807

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 6788	. 2 |- X_ x e. A B = { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
2		nfcv 2348	. . . . 5 |- F/_ x y
3		nfab1 2350	. . . . 5 |- F/_ x { x | x e. A }
4	2, 3	nffn 5371	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2537	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1588	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2353	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2345	1 |- F/_ x X_ x e. A B

Syntax hints:   /\ wa 104   e. wcel 2176   {cab 2191   F/_wnfc 2335   A.wral 2484   Fn wfn 5267   ` cfv 5272   X_cixp 6787

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-fun 5274  df-fn 5275  df-ixp 6788

This theorem is referenced by: (None)