Theorem ixp0x 7057

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 7032	. 2 |- X_ x e. (/) A = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
2		elsn 3797	. . . 4 |- ( f e. { (/) } <-> f = (/) )
3		fn0 5531	. . . 4 |- ( f Fn (/) <-> f = (/) )
4		ral0 3700	. . . . 5 |- A. x e. (/) ( f ` x ) e. A
5	4	biantru 492	. . . 4 |- ( f Fn (/) <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
6	2, 3, 5	3bitr2i 265	. . 3 |- ( f e. { (/) } <-> ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) )
7	6	abbi2i 2523	. 2 |- { (/) } = { f | ( f Fn (/) /\ A. x e. (/) ( f ` x ) e. A ) }
8	1, 7	eqtr4i 2435	1 |- X_ x e. (/) A = { (/) }

Syntax hints:   /\ wa 359   = wceq 1649   e. wcel 1721   {cab 2398   A.wral 2674   (/)c0 3596   {csn 3782   Fn wfn 5416   ` cfv 5421   X_cixp 7030

This theorem is referenced by:  0elixp  7060  ptcmpfi  17806

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-fun 5423  df-fn 5424  df-ixp 7031