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Theorem isufil 21927
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 21925 . 2 UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧)})
2 pweq 4301 . . . 4 (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
32adantr 466 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋)
4 eleq2 2839 . . . . 5 (𝑧 = 𝐹 → (𝑥𝑧𝑥𝐹))
54adantl 467 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → (𝑥𝑧𝑥𝐹))
6 difeq1 3872 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑥) = (𝑋𝑥))
7 eleq12 2840 . . . . 5 (((𝑦𝑥) = (𝑋𝑥) ∧ 𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
86, 7sylan 569 . . . 4 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑦𝑥) ∈ 𝑧 ↔ (𝑋𝑥) ∈ 𝐹))
95, 8orbi12d 904 . . 3 ((𝑦 = 𝑋𝑧 = 𝐹) → ((𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
103, 9raleqbidv 3301 . 2 ((𝑦 = 𝑋𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥𝑧 ∨ (𝑦𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
11 fveq2 6333 . 2 (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋))
12 fvex 6344 . 2 (Fil‘𝑦) ∈ V
13 elfvdm 6363 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
141, 10, 11, 12, 13elmptrab2 21852 1 (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 836   = wceq 1631  wcel 2145  wral 3061  cdif 3720  𝒫 cpw 4298  dom cdm 5250  cfv 6030  Filcfil 21869  UFilcufil 21923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ufil 21925
This theorem is referenced by:  ufilfil  21928  ufilss  21929  isufil2  21932  trufil  21934  fixufil  21946  fin1aufil  21956
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