Theorem isufil 23806

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 23804	. 2 ⊢ UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧)})
2		pweq 4567	. . . 4 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
3	2	adantr 480	. . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4		eleq2 2817	. . . . 5 ⊢ (𝑧 = 𝐹 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
5	4	adantl 481	. . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹))
6		difeq1 4072	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥))
7		eleq12 2818	. . . . 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 3310	. 2 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
11		fveq2 6826	. 2 ⊢ (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12		fvex 6839	. 2 ⊢ (Fil‘𝑦) ∈ V
13		elfvdm 6861	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
14	1, 10, 11, 12, 13	elmptrab2 23731	1 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3902  𝒫 cpw 4553  dom cdm 5623  ‘cfv 6486  Filcfil 23748  UFilcufil 23802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ufil 23804

This theorem is referenced by:  ufilfil  23807  ufilss  23808  isufil2  23811  trufil  23813  fixufil  23825  fin1aufil  23835