Theorem nfixpxy 6806

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 6788	. 2 |- X_ x e. A B = { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
2		nfcv 2348	. . . . 5 |- F/_ y z
3		nftru 1489	. . . . . . 7 |- F/ x T.
4		nfcvd 2349	. . . . . . . 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 2364	. . . . . . 7 |- ( T. -> F/ y x e. A )
8	3, 7	nfabd 2368	. . . . . 6 |- ( T. -> F/_ y { x | x e. A } )
9	8	mptru 1382	. . . . 5 |- F/_ y { x | x e. A }
10	2, 9	nffn 5371	. . . 4 |- F/ y z Fn { x | x e. A }
11		df-ral 2489	. . . . 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 5590	. . . . . . . . 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 2364	. . . . . . . 8 |- ( T. -> F/ y ( z ` x ) e. B )
17	7, 16	nfimd 1608	. . . . . . 7 |- ( T. -> F/ y ( x e. A -> ( z ` x ) e. B ) )
18	3, 17	nfald 1783	. . . . . 6 |- ( T. -> F/ y A. x ( x e. A -> ( z ` x ) e. B ) )
19	18	mptru 1382	. . . . 5 |- F/ y A. x ( x e. A -> ( z ` x ) e. B )
20	11, 19	nfxfr 1497	. . . 4 |- F/ y A. x e. A ( z ` x ) e. B
21	10, 20	nfan 1588	. . 3 |- F/ y ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B )
22	21	nfab 2353	. 2 |- F/_ y { z | ( z Fn { x | x e. A } /\ A. x e. A ( z ` x ) e. B ) }
23	1, 22	nfcxfr 2345	1 |- F/_ y X_ x e. A B

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   T. wtru 1374   F/wnf 1483   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-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ixp 6788

This theorem is referenced by: (None)