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Theorem isufil 23898
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 23896 . 2 UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧)})
2 pweq 4621 . . . 4 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
32adantr 479 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4 eleq2 2815 . . . . 5 (𝑧 = 𝐹 → (𝑥𝑧𝑥𝐹))
54adantl 480 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → (𝑥𝑧𝑥𝐹))
6 difeq1 4114 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑥) = (𝑋𝑥))
7 eleq12 2816 . . . . 5 (((𝑦𝑥) = (𝑋𝑥) ∧ 𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
86, 7sylan 578 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
95, 8orbi12d 916 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
103, 9raleqbidv 3330 . 2 ((𝑦 = 𝑋𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
11 fveq2 6901 . 2 (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12 fvex 6914 . 2 (Fil‘𝑦) ∈ V
13 elfvdm 6938 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
141, 10, 11, 12, 13elmptrab2 23823 1 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wo 845   = wceq 1534  wcel 2099  wral 3051  cdif 3944  𝒫 cpw 4607  dom cdm 5682  cfv 6554  Filcfil 23840  UFilcufil 23894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fv 6562  df-ufil 23896
This theorem is referenced by:  ufilfil  23899  ufilss  23900  isufil2  23903  trufil  23905  fixufil  23917  fin1aufil  23927
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