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Theorem isufil 22506
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 22504 . 2 UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧)})
2 pweq 4527 . . . 4 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
32adantr 484 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4 eleq2 2902 . . . . 5 (𝑧 = 𝐹 → (𝑥𝑧𝑥𝐹))
54adantl 485 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → (𝑥𝑧𝑥𝐹))
6 difeq1 4067 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑥) = (𝑋𝑥))
7 eleq12 2903 . . . . 5 (((𝑦𝑥) = (𝑋𝑥) ∧ 𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
86, 7sylan 583 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
95, 8orbi12d 916 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
103, 9raleqbidv 3382 . 2 ((𝑦 = 𝑋𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
11 fveq2 6652 . 2 (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12 fvex 6665 . 2 (Fil‘𝑦) ∈ V
13 elfvdm 6684 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
141, 10, 11, 12, 13elmptrab2 22431 1 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844   = wceq 1538  wcel 2114  wral 3130  cdif 3905  𝒫 cpw 4511  dom cdm 5532  cfv 6334  Filcfil 22448  UFilcufil 22502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-ufil 22504
This theorem is referenced by:  ufilfil  22507  ufilss  22508  isufil2  22511  trufil  22513  fixufil  22525  fin1aufil  22535
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