Theorem fi0 9335

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 5254	. . 3 ⊢ ∅ ∈ V
2		fival 9327	. . 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 5262	. . . 4 ⊢ ¬ V ∈ V
5		id 22	. . . . . . 7 ⊢ (𝑦 = ∩ 𝑥 → 𝑦 = ∩ 𝑥)
6		elinel1 4155	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 ∈ 𝒫 ∅)
7		elpwi 4563	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ∅ → 𝑥 ⊆ ∅)
8		ss0 4356	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ∅ → 𝑥 = ∅)
9	6, 7, 8	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → 𝑥 = ∅)
10	9	inteqd 4909	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = ∩ ∅)
11		int0 4919	. . . . . . . 8 ⊢ ∩ ∅ = V
12	10, 11	eqtrdi 2788	. . . . . . 7 ⊢ (𝑥 ∈ (𝒫 ∅ ∩ Fin) → ∩ 𝑥 = V)
13	5, 12	sylan9eqr 2794	. . . . . 6 ⊢ ((𝑥 ∈ (𝒫 ∅ ∩ Fin) ∧ 𝑦 = ∩ 𝑥) → 𝑦 = V)
14	13	rexlimiva 3131	. . . . 5 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → 𝑦 = V)
15		vex 3446	. . . . 5 ⊢ 𝑦 ∈ V
16	14, 15	eqeltrrdi 2846	. . . 4 ⊢ (∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥 → V ∈ V)
17	4, 16	mto 197	. . 3 ⊢ ¬ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥
18	17	abf 4360	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ (𝒫 ∅ ∩ Fin)𝑦 = ∩ 𝑥} = ∅
19	3, 18	eqtri 2760	1 ⊢ (fi‘∅) = ∅

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∩ cint 4904  ‘cfv 6500  Fincfn 8895  ficfi 9325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-fi 9326

This theorem is referenced by:  fieq0  9336  firest  17364  restbas  23114