Theorem fi0 7940

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 4557	. . 3 |- (/) e. _V
2		fival 7932	. . 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 4563	. . . 4 |- -. _V e. _V
5		id 23	. . . . . . 7 |- ( y = |^| x -> y = |^| x )
6		inss1 3688	. . . . . . . . . . 11 |- ( ~P (/) i^i Fin ) C_ ~P (/)
7	6	sseli 3466	. . . . . . . . . 10 |- ( x e. ( ~P (/) i^i Fin ) -> x e. ~P (/) )
8		elpwi 3994	. . . . . . . . . 10 |- ( x e. ~P (/) -> x C_ (/) )
9		ss0 3799	. . . . . . . . . 10 |- ( x C_ (/) -> x = (/) )
10	7, 8, 9	3syl 18	. . . . . . . . 9 |- ( x e. ( ~P (/) i^i Fin ) -> x = (/) )
11	10	inteqd 4263	. . . . . . . 8 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = |^| (/) )
12		int0 4272	. . . . . . . 8 |- |^| (/) = _V
13	11, 12	syl6eq 2486	. . . . . . 7 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = _V )
14	5, 13	sylan9eqr 2492	. . . . . 6 |- ( ( x e. ( ~P (/) i^i Fin ) /\ y = |^| x ) -> y = _V )
15	14	rexlimiva 2920	. . . . 5 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> y = _V )
16		vex 3090	. . . . 5 |- y e. _V
17	15, 16	syl6eqelr 2526	. . . 4 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> _V e. _V )
18	4, 17	mto 179	. . 3 |- -. E. x e. ( ~P (/) i^i Fin ) y = |^| x
19	18	abf 3802	. 2 |- { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } = (/)
20	3, 19	eqtri 2458	1 |- ( fi ` (/) ) = (/)

Syntax hints:   = wceq 1437   e. wcel 1870   {cab 2414   E.wrex 2783   _Vcvv 3087   i^i cin 3441   C_ wss 3442   (/)c0 3767   ~Pcpw 3985   |^|cint 4258   ` cfv 5601   Fincfn 7577   ficfi 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-fi 7931

This theorem is referenced by:  fieq0  7941  firest  15290  restbas  20105