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Mirrors > Home > MPE Home > Th. List > isufil | Structured version Visualization version GIF version |
Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.) |
Ref | Expression |
---|---|
isufil | ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ufil 22124 | . 2 ⊢ UFil = (𝑦 ∈ V ↦ {𝑧 ∈ (Fil‘𝑦) ∣ ∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧)}) | |
2 | pweq 4382 | . . . 4 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋) | |
3 | 2 | adantr 474 | . . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → 𝒫 𝑦 = 𝒫 𝑋) |
4 | eleq2 2848 | . . . . 5 ⊢ (𝑧 = 𝐹 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹)) | |
5 | 4 | adantl 475 | . . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐹)) |
6 | difeq1 3944 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥)) | |
7 | eleq12 2849 | . . . . 5 ⊢ (((𝑦 ∖ 𝑥) = (𝑋 ∖ 𝑥) ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹)) | |
8 | 6, 7 | sylan 575 | . . . 4 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑦 ∖ 𝑥) ∈ 𝑧 ↔ (𝑋 ∖ 𝑥) ∈ 𝐹)) |
9 | 5, 8 | orbi12d 905 | . . 3 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → ((𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
10 | 3, 9 | raleqbidv 3326 | . 2 ⊢ ((𝑦 = 𝑋 ∧ 𝑧 = 𝐹) → (∀𝑥 ∈ 𝒫 𝑦(𝑥 ∈ 𝑧 ∨ (𝑦 ∖ 𝑥) ∈ 𝑧) ↔ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
11 | fveq2 6448 | . 2 ⊢ (𝑦 = 𝑋 → (Fil‘𝑦) = (Fil‘𝑋)) | |
12 | fvex 6461 | . 2 ⊢ (Fil‘𝑦) ∈ V | |
13 | elfvdm 6480 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil) | |
14 | 1, 10, 11, 12, 13 | elmptrab2 22051 | 1 ⊢ (𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∖ cdif 3789 𝒫 cpw 4379 dom cdm 5357 ‘cfv 6137 Filcfil 22068 UFilcufil 22122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fv 6145 df-ufil 22124 |
This theorem is referenced by: ufilfil 22127 ufilss 22128 isufil2 22131 trufil 22133 fixufil 22145 fin1aufil 22155 |
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