Theorem ixp0x 7093

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 7068	. 2 |- X_ x e. (/) A = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
2		elsn 3831	. . . 4 |- ( f e. { (/) } <-> f = (/) )
3		fn0 5567	. . . 4 |- ( f Fn (/) <-> f = (/) )
4		ral0 3734	. . . . 5 |- A. x e. (/) ( f ` x ) e. A
5	4	biantru 493	. . . 4 |- ( f Fn (/) <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
6	2, 3, 5	3bitr2i 266	. . 3 |- ( f e. { (/) } <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
7	6	abbi2i 2549	. 2 |- { (/) } = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
8	1, 7	eqtr4i 2461	1 |- X_ x e. (/) A = { (/) }

Syntax hints:   /\ wa 360   = wceq 1653   e. wcel 1726   {cab 2424   A.wral 2707   (/)c0 3630   {csn 3816   Fn wfn 5452   ` cfv 5457   X_cixp 7066

This theorem is referenced by:  0elixp  7096  ptcmpfi  17850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-fun 5459  df-fn 5460  df-ixp 7067