Theorem isufil 22962

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 22960	. 2 ⊢ UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧)})
2		pweq 4546	. . . 4 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
3	2	adantr 480	. . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4		eleq2 2827	. . . . 5 ⊢ (𝑧 = 𝐹 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
5	4	adantl 481	. . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
6		difeq1 4046	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥))
7		eleq12 2828	. . . . 5 ⊢ (((𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
8	6, 7	sylan 579	. . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹))
9	5, 8	orbi12d 915	. . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
10	3, 9	raleqbidv 3327	. 2 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
11		fveq2 6756	. 2 ⊢ (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12		fvex 6769	. 2 ⊢ (Fil‘𝑦) ∈ V
13		elfvdm 6788	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
14	1, 10, 11, 12, 13	elmptrab2 22887	1 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∖ cdif 3880  𝒫 cpw 4530  dom cdm 5580  ‘cfv 6418  Filcfil 22904  UFilcufil 22958

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-pr 5347

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-sbc 3712  df-csb 3829  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-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-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fv 6426  df-ufil 22960

This theorem is referenced by:  ufilfil  22963  ufilss  22964  isufil2  22967  trufil  22969  fixufil  22981  fin1aufil  22991