Theorem ixp0x 7936

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 7910	. 2 |- X_ x e. (/) A = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
2		velsn 4193	. . . 4 |- ( f e. { (/) } <-> f = (/) )
3		fn0 6011	. . . 4 |- ( f Fn (/) <-> f = (/) )
4		ral0 4076	. . . . 5 |- A. x e. (/) ( f ` x ) e. A
5	4	biantru 526	. . . 4 |- ( f Fn (/) <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
6	2, 3, 5	3bitr2i 288	. . 3 |- ( f e. { (/) } <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
7	6	abbi2i 2738	. 2 |- { (/) } = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
8	1, 7	eqtr4i 2647	1 |- X_ x e. (/) A = { (/) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   (/)c0 3915   {csn 4177   Fn wfn 5883   ` cfv 5888   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891  df-ixp 7909

This theorem is referenced by:  0elixp  7939  ptcmpfi  21616  finixpnum  33394  ioorrnopn  40525  ioorrnopnxr  40527  hoicvr  40762  ovnhoi  40817  ovnlecvr2  40824  hoiqssbl  40839  hoimbl  40845  iunhoiioo  40890