Theorem nfixpxy 6683

Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)

Hypotheses

Ref	Expression
nfixp.1	|- F/_ y A
nfixp.2	|- F/_ y B

Assertion

Ref	Expression
nfixpxy	|- F/_ y X_ x e. A B

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem nfixpxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ixp 6665	. 2 |- X_ x e. A B = { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
2		nfcv 2308	. . . . 5 |- F/_ y z
3		nftru 1454	. . . . . . 7 |- F/ x T.
4		nfcvd 2309	. . . . . . . 8 |- ( T. -> F/_ y x )
5		nfixp.1	. . . . . . . . 9 |- F/_ y A
6	5	a1i 9	. . . . . . . 8 |- ( T. -> F/_ y A )
7	4, 6	nfeld 2324	. . . . . . 7 |- ( T. -> F/ y x e. A )
8	3, 7	nfabd 2328	. . . . . 6 |- ( T. -> F/_ y { x | x e. A } )
9	8	mptru 1352	. . . . 5 |- F/_ y { x | x e. A }
10	2, 9	nffn 5284	. . . 4 |- F/ y z Fn { x | x e. A }
11		df-ral 2449	. . . . 5 |- ( A. x e. A ( z ` x ) e. B <-> A. x ( x e. A -> ( z ` x ) e. B ) )
12	2	a1i 9	. . . . . . . . . 10 |- ( T. -> F/_ y z )
13	12, 4	nffvd 5498	. . . . . . . . 9 |- ( T. -> F/_ y ( z ` x ) )
14		nfixp.2	. . . . . . . . . 10 |- F/_ y B
15	14	a1i 9	. . . . . . . . 9 |- ( T. -> F/_ y B )
16	13, 15	nfeld 2324	. . . . . . . 8 |- ( T. -> F/ y ( z ` x ) e. B )
17	7, 16	nfimd 1573	. . . . . . 7 |- ( T. -> F/ y ( x e. A -> ( z ` x ) e. B ) )
18	3, 17	nfald 1748	. . . . . 6 |- ( T. -> F/ y A. x ( x e. A -> ( z ` x ) e. B ) )
19	18	mptru 1352	. . . . 5 |- F/ y A. x ( x e. A -> ( z ` x ) e. B )
20	11, 19	nfxfr 1462	. . . 4 |- F/ y A. x e. A ( z ` x ) e. B
21	10, 20	nfan 1553	. . 3 |- F/ y ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B )
22	21	nfab 2313	. 2 |- F/_ y { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
23	1, 22	nfcxfr 2305	1 |- F/_ y X_ x e. A B

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   T. wtru 1344   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   A.wral 2444   Fn wfn 5183   ` cfv 5188   X_cixp 6664

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-ixp 6665

This theorem is referenced by: (None)