Theorem ixp0x 6720

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 6694	. 2 |- X_ x e. (/) A = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
2		velsn 3608	. . . 4 |- ( f e. { (/) } <-> f = (/) )
3		fn0 5331	. . . 4 |- ( f Fn (/) <-> f = (/) )
4		ral0 3524	. . . . 5 |- A. x e. (/) ( f ` x ) e. A
5	4	biantru 302	. . . 4 |- ( f Fn (/) <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
6	2, 3, 5	3bitr2i 208	. . 3 |- ( f e. { (/) } <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
7	6	abbi2i 2292	. 2 |- { (/) } = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
8	1, 7	eqtr4i 2201	1 |- X_ x e. (/) A = { (/) }

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   (/)c0 3422   {csn 3591   Fn wfn 5207   ` cfv 5212   X_cixp 6692

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-fun 5214  df-fn 5215  df-ixp 6693

This theorem is referenced by:  0elixp  6723