Theorem ixp0x 6550

Description: An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.)

Assertion

Ref	Expression
ixp0x	|- X_ x e. (/) A = { (/) }

Proof of Theorem ixp0x

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfixp 6524	. 2 |- X_ x e. (/) A = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
2		velsn 3491	. . . 4 |- ( f e. { (/) } <-> f = (/) )
3		fn0 5178	. . . 4 |- ( f Fn (/) <-> f = (/) )
4		ral0 3411	. . . . 5 |- A. x e. (/) ( f ` x ) e. A
5	4	biantru 298	. . . 4 |- ( f Fn (/) <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
6	2, 3, 5	3bitr2i 207	. . 3 |- ( f e. { (/) } <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
7	6	abbi2i 2214	. 2 |- { (/) } = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
8	1, 7	eqtr4i 2123	1 |- X_ x e. (/) A = { (/) }

Syntax hints:   /\ wa 103   = wceq 1299   e. wcel 1448   {cab 2086   A.wral 2375   (/)c0 3310   {csn 3474   Fn wfn 5054   ` cfv 5059   X_cixp 6522

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-fun 5061  df-fn 5062  df-ixp 6523

This theorem is referenced by:  0elixp  6553