Theorem fi0 7952

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 4528	. . 3 |- (/) e. _V
2		fival 7944	. . 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 4534	. . . 4 |- -. _V e. _V
5		id 22	. . . . . . 7 |- ( y = |^| x -> y = |^| x )
6		inss1 3643	. . . . . . . . . . 11 |- ( ~P (/) i^i Fin ) C_ ~P (/)
7	6	sseli 3414	. . . . . . . . . 10 |- ( x e. ( ~P (/) i^i Fin ) -> x e. ~P (/) )
8		elpwi 3951	. . . . . . . . . 10 |- ( x e. ~P (/) -> x C_ (/) )
9		ss0 3768	. . . . . . . . . 10 |- ( x C_ (/) -> x = (/) )
10	7, 8, 9	3syl 18	. . . . . . . . 9 |- ( x e. ( ~P (/) i^i Fin ) -> x = (/) )
11	10	inteqd 4231	. . . . . . . 8 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = |^| (/) )
12		int0 4240	. . . . . . . 8 |- |^| (/) = _V
13	11, 12	syl6eq 2521	. . . . . . 7 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = _V )
14	5, 13	sylan9eqr 2527	. . . . . 6 |- ( ( x e. ( ~P (/) i^i Fin ) /\ y = |^| x ) -> y = _V )
15	14	rexlimiva 2868	. . . . 5 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> y = _V )
16		vex 3034	. . . . 5 |- y e. _V
17	15, 16	syl6eqelr 2558	. . . 4 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> _V e. _V )
18	4, 17	mto 181	. . 3 |- -. E. x e. ( ~P (/) i^i Fin ) y = |^| x
19	18	abf 3772	. 2 |- { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } = (/)
20	3, 19	eqtri 2493	1 |- ( fi ` (/) ) = (/)

Syntax hints:   = wceq 1452   e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   |^|cint 4226   ` cfv 5589   Fincfn 7587   ficfi 7942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-fi 7943

This theorem is referenced by:  fieq0  7953  firest  15409  restbas  20251