Theorem nfixpxy 6929

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 6911	. 2 |- X_ x e. A B = { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
2		nfcv 2375	. . . . 5 |- F/_ y z
3		nftru 1515	. . . . . . 7 |- F/ x T.
4		nfcvd 2376	. . . . . . . 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 2391	. . . . . . 7 |- ( T. -> F/ y x e. A )
8	3, 7	nfabd 2395	. . . . . 6 |- ( T. -> F/_ y { x | x e. A } )
9	8	mptru 1407	. . . . 5 |- F/_ y { x | x e. A }
10	2, 9	nffn 5433	. . . 4 |- F/ y z Fn { x | x e. A }
11		df-ral 2516	. . . . 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 5660	. . . . . . . . 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 2391	. . . . . . . 8 |- ( T. -> F/ y ( z ` x ) e. B )
17	7, 16	nfimd 1634	. . . . . . 7 |- ( T. -> F/ y ( x e. A -> ( z ` x ) e. B ) )
18	3, 17	nfald 1808	. . . . . 6 |- ( T. -> F/ y A. x ( x e. A -> ( z ` x ) e. B ) )
19	18	mptru 1407	. . . . 5 |- F/ y A. x ( x e. A -> ( z ` x ) e. B )
20	11, 19	nfxfr 1523	. . . 4 |- F/ y A. x e. A ( z ` x ) e. B
21	10, 20	nfan 1614	. . 3 |- F/ y ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B )
22	21	nfab 2380	. 2 |- F/_ y { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
23	1, 22	nfcxfr 2372	1 |- F/_ y X_ x e. A B

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   T. wtru 1399   F/wnf 1509   e. wcel 2202   {cab 2217   F/_wnfc 2362   A.wral 2511   Fn wfn 5328   ` cfv 5333   X_cixp 6910

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ixp 6911

This theorem is referenced by: (None)