Theorem nfixpxy 6785

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 6767	. 2 |- X_ x e. A B = { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
2		nfcv 2339	. . . . 5 |- F/_ y z
3		nftru 1480	. . . . . . 7 |- F/ x T.
4		nfcvd 2340	. . . . . . . 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 2355	. . . . . . 7 |- ( T. -> F/ y x e. A )
8	3, 7	nfabd 2359	. . . . . 6 |- ( T. -> F/_ y { x | x e. A } )
9	8	mptru 1373	. . . . 5 |- F/_ y { x | x e. A }
10	2, 9	nffn 5355	. . . 4 |- F/ y z Fn { x | x e. A }
11		df-ral 2480	. . . . 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 5573	. . . . . . . . 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 2355	. . . . . . . 8 |- ( T. -> F/ y ( z ` x ) e. B )
17	7, 16	nfimd 1599	. . . . . . 7 |- ( T. -> F/ y ( x e. A -> ( z ` x ) e. B ) )
18	3, 17	nfald 1774	. . . . . 6 |- ( T. -> F/ y A. x ( x e. A -> ( z ` x ) e. B ) )
19	18	mptru 1373	. . . . 5 |- F/ y A. x ( x e. A -> ( z ` x ) e. B )
20	11, 19	nfxfr 1488	. . . 4 |- F/ y A. x e. A ( z ` x ) e. B
21	10, 20	nfan 1579	. . 3 |- F/ y ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B )
22	21	nfab 2344	. 2 |- F/_ y { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
23	1, 22	nfcxfr 2336	1 |- F/_ y X_ x e. A B

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   T. wtru 1365   F/wnf 1474   e. wcel 2167   {cab 2182   F/_wnfc 2326   A.wral 2475   Fn wfn 5254   ` cfv 5259   X_cixp 6766

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ixp 6767

This theorem is referenced by: (None)