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Theorem isufil 23911
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 23909 . 2 UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧)})
2 pweq 4614 . . . 4 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
32adantr 480 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4 eleq2 2830 . . . . 5 (𝑧 = 𝐹 → (𝑥𝑧𝑥𝐹))
54adantl 481 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → (𝑥𝑧𝑥𝐹))
6 difeq1 4119 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑥) = (𝑋𝑥))
7 eleq12 2831 . . . . 5 (((𝑦𝑥) = (𝑋𝑥) ∧ 𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
86, 7sylan 580 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
95, 8orbi12d 919 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
103, 9raleqbidv 3346 . 2 ((𝑦 = 𝑋𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
11 fveq2 6906 . 2 (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12 fvex 6919 . 2 (Fil‘𝑦) ∈ V
13 elfvdm 6943 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
141, 10, 11, 12, 13elmptrab2 23836 1 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cdif 3948  𝒫 cpw 4600  dom cdm 5685  cfv 6561  Filcfil 23853  UFilcufil 23907
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ufil 23909
This theorem is referenced by:  ufilfil  23912  ufilss  23913  isufil2  23916  trufil  23918  fixufil  23930  fin1aufil  23940
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