Theorem isufil 23819

Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)

Assertion

Ref	Expression
isufil	⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋

Proof of Theorem isufil

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ufil 23817	. 2 ⊢ UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧)})
2		pweq 4564	. . . 4 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
3	2	adantr 480	. . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4		eleq2 2820	. . . . 5 ⊢ (𝑧 = 𝐹 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
5	4	adantl 481	. . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
6		difeq1 4069	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥))
7		eleq12 2821	. . . . 5 ⊢ (((𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
8	6, 7	sylan 580	. . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
9	5, 8	orbi12d 918	. . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
10	3, 9	raleqbidv 3312	. 2 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
11		fveq2 6822	. 2 ⊢ (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12		fvex 6835	. 2 ⊢ (Fil‘𝑦) ∈ V
13		elfvdm 6856	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
14	1, 10, 11, 12, 13	elmptrab2 23744	1 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∖ cdif 3899  𝒫 cpw 4550  dom cdm 5616  ‘cfv 6481  Filcfil 23761  UFilcufil 23815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ufil 23817

This theorem is referenced by:  ufilfil  23820  ufilss  23821  isufil2  23824  trufil  23826  fixufil  23838  fin1aufil  23848