Theorem fi0 8326

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 4790	. . 3 |- (/) e. _V
2		fival 8318	. . 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 4796	. . . 4 |- -. _V e. _V
5		id 22	. . . . . . 7 |- ( y = |^| x -> y = |^| x )
6		inss1 3833	. . . . . . . . . . 11 |- ( ~P (/) i^i Fin ) C_ ~P (/)
7	6	sseli 3599	. . . . . . . . . 10 |- ( x e. ( ~P (/) i^i Fin ) -> x e. ~P (/) )
8		elpwi 4168	. . . . . . . . . 10 |- ( x e. ~P (/) -> x C_ (/) )
9		ss0 3974	. . . . . . . . . 10 |- ( x C_ (/) -> x = (/) )
10	7, 8, 9	3syl 18	. . . . . . . . 9 |- ( x e. ( ~P (/) i^i Fin ) -> x = (/) )
11	10	inteqd 4480	. . . . . . . 8 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = |^| (/) )
12		int0 4490	. . . . . . . 8 |- |^| (/) = _V
13	11, 12	syl6eq 2672	. . . . . . 7 |- ( x e. ( ~P (/) i^i Fin ) -> |^| x = _V )
14	5, 13	sylan9eqr 2678	. . . . . 6 |- ( ( x e. ( ~P (/) i^i Fin ) /\ y = |^| x ) -> y = _V )
15	14	rexlimiva 3028	. . . . 5 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> y = _V )
16		vex 3203	. . . . 5 |- y e. _V
17	15, 16	syl6eqelr 2710	. . . 4 |- ( E. x e. ( ~P (/) i^i Fin ) y = |^| x -> _V e. _V )
18	4, 17	mto 188	. . 3 |- -. E. x e. ( ~P (/) i^i Fin ) y = |^| x
19	18	abf 3978	. 2 |- { y | E. x e. ( ~P (/) i^i Fin ) y = |^| x } = (/)
20	3, 19	eqtri 2644	1 |- ( fi ` (/) ) = (/)

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` cfv 5888   Fincfn 7955   ficfi 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-fi 8317

This theorem is referenced by:  fieq0  8327  firest  16093  restbas  20962