Theorem fi0 9109

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 5226	. . 3 ⊢ ∅ ∈ V
2		fival 9101	. . 3 ⊢ (∅ ∈ V → (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥})
3	1, 2	ax-mp 5	. 2 ⊢ (fi‘∅) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥}
4		vprc 5234	. . . 4 ⊢ ¬ V ∈ V
5		id 22	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥)
6		elinel1 4125	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
7		elpwi 4539	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
8		ss0 4329	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
9	6, 7, 8	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
10	9	inteqd 4881	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅)
11		int0 4890	. . . . . . . 8 ⊢ ∩ ∅ = V
12	10, 11	eqtrdi 2795	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V)
13	5, 12	sylan9eqr 2801	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V)
14	13	rexlimiva 3209	. . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V)
15		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
16	14, 15	eqeltrrdi 2848	. . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V)
17	4, 16	mto 196	. . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥
18	17	abf 4333	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅
19	3, 18	eqtri 2766	1 ⊢ (fi‘∅) = ∅

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∩ cint 4876  ‘cfv 6418  Fincfn 8691  ficfi 9099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-fi 9100

This theorem is referenced by:  fieq0  9110  firest  17060  restbas  22217