Theorem fi0 7880

Description: The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
fi0	|- ( fi ` (/) ) = (/)

Proof of Theorem fi0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4577	. . 3 |- (/) e. _V
2		fival 7872	. . 3 |- ( (/) e. _V -> ( fi ` (/) ) = { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } )
3	1, 2	ax-mp 5	. 2 |- ( fi ` (/) ) = { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x }
4		vprc 4585	. . . 4 |- -. _V e. _V
5		id 22	. . . . . . 7 |- ( y = |^| x -> y = |^| x )
6		inss1 3718	. . . . . . . . . . 11 |- ( ~P (/) i^i Fin ) C_ ~P (/)
7	6	sseli 3500	. . . . . . . . . 10 |- ( x e. ( ~P (/) i^i Fin ) -> x e. ~P (/) )
8		elpwi 4019	. . . . . . . . . 10 |- ( x e. ~P (/) -> x C_ (/) )
9		ss0 3816	. . . . . . . . . 10 |- ( x C_ (/) -> x = (/) )
10	7, 8, 9	3syl 20	. . . . . . . . 9 |- ( x e. ( ~P (/) i^i Fin ) -> x = (/) )
11	10	inteqd 4287	. . . . . . . 8 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = |^| (/) )
12		int0 4296	. . . . . . . 8 |- |^| (/) = _V
13	11, 12	syl6eq 2524	. . . . . . 7 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = _V )
14	5, 13	sylan9eqr 2530	. . . . . 6 |- ( ( x e. ( ~P (/) i^i Fin ) /\ y = |^| x ) -> y = _V )
15	14	rexlimiva 2951	. . . . 5 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> y = _V )
16		vex 3116	. . . . 5 |- y e. _V
17	15, 16	syl6eqelr 2564	. . . 4 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> _V e. _V )
18	4, 17	mto 176	. . 3 |- -. E. x e. ( ~P (/) i^i Fin ) y = |^| x
19	18	abf 3819	. 2 |- { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } = (/)
20	3, 19	eqtri 2496	1 |- ( fi ` (/) ) = (/)

Syntax hints:   = wceq 1379   e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |^|cint 4282   ` cfv 5588   Fincfn 7516   ficfi 7870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-fi 7871

This theorem is referenced by:  fieq0  7881  firest  14688  restbas  19453