Theorem nfixp1 7928

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 7909	. 2 |- X_ x e. A B = { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
2		nfcv 2764	. . . . 5 |- F/_ x y
3		nfab1 2766	. . . . 5 |- F/_ x { x | x e. A }
4	2, 3	nffn 5987	. . . 4 |- F/ x y Fn { x | x e. A }
5		nfra1 2941	. . . 4 |- F/ x A. x e. A ( y ` x ) e. B
6	4, 5	nfan 1828	. . 3 |- F/ x ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B )
7	6	nfab 2769	. 2 |- F/_ x { y | ( y Fn { x | x e. A } /\ A. x e. A ( y ` x ) e. B ) }
8	1, 7	nfcxfr 2762	1 |- F/_ x X_ x e. A B

Syntax hints:   /\ wa 384   e. wcel 1990   {cab 2608   F/_wnfc 2751   A.wral 2912   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891  df-ixp 7909

This theorem is referenced by:  ixpiunwdom  8496  ptbasfi  21384  hoidmvlelem3  40811  hspdifhsp  40830  hoiqssbllem2  40837  hspmbllem2  40841  opnvonmbllem2  40847  iinhoiicc  40888  iunhoiioo  40890  vonioo  40896  vonicc  40899