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Theorem isufil 21517
 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(𝑥𝐹 ∨ (𝑋𝑥) ∈ 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.
StepHypRef Expression
1 df-ufil 21515 . 2 UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧)})
2 pweq 4111 . . . 4 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
32adantr 480 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4 eleq2 2677 . . . . 5 (𝑧 = 𝐹 → (𝑥𝑧𝑥𝐹))
54adantl 481 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → (𝑥𝑧𝑥𝐹))
6 difeq1 3683 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑥) = (𝑋𝑥))
7 eleq12 2678 . . . . 5 (((𝑦𝑥) = (𝑋𝑥) ∧ 𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
86, 7sylan 487 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
95, 8orbi12d 742 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
103, 9raleqbidv 3129 . 2 ((𝑦 = 𝑋𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
11 fveq2 6103 . 2 (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12 fvex 6113 . 2 (Fil‘𝑦) ∈ V
13 elfvdm 6130 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
141, 10, 11, 12, 13elmptrab2 21442 1 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  𝒫 cpw 4108  dom cdm 5038  ‘cfv 5804  Filcfil 21459  UFilcufil 21513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ufil 21515 This theorem is referenced by:  ufilfil  21518  ufilss  21519  isufil2  21522  trufil  21524  fixufil  21536  fin1aufil  21546
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