Theorem fixufil 23310

Description: The condition describing a fixed ultrafilter always produces an ultrafilter. (Contributed by Jeff Hankins, 9-Dec-2009.) (Revised by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 29-Jul-2015.)

Assertion

Ref	Expression
fixufil	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑉

Proof of Theorem fixufil

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uffix 23309	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
2	1	simprd 496	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
3	1	simpld 495	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4		fgcl 23266	. . . 4 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
5	3, 4	syl 17	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
6	2, 5	eqeltrd 2832	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
7		undif2 4441	. . . . . . . . . 10 ⊢ (𝑦 ∪ (𝑋 ∖ 𝑦)) = (𝑦 ∪ 𝑋)
8		elpwi 4572	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9		ssequn1 4145	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∪ 𝑋) = 𝑋)
10	8, 9	sylib 217	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝑦 ∪ 𝑋) = 𝑋)
11	7, 10	eqtr2id 2784	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑋 = (𝑦 ∪ (𝑋 ∖ 𝑦)))
12	11	eleq2d 2818	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦))))
13	12	biimpac 479	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)))
14		elun 4113	. . . . . . 7 ⊢ (𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
15	13, 14	sylib 217	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
16	15	adantll 712	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
17		ibar 529	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
18	17	adantl 482	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
19		difss 4096	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝑦) ⊆ 𝑋
20		elpw2g 5306	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
21	19, 20	mpbiri 257	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
22	21	ad2antrr 724	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
23	22	biantrurd 533	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋 ∖ 𝑦) ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
24	18, 23	orbi12d 917	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))))
25	16, 24	mpbid 231	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
26	25	ralrimiva 3139	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
27		eleq2 2821	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
28	27	elrab 3648	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦))
29		eleq2 2821	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (𝑋 ∖ 𝑦)))
30	29	elrab 3648	. . . . 5 ⊢ ((𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))
31	28, 30	orbi12i 913	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
32	31	ralbii 3092	. . 3 ⊢ (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
33	26, 32	sylibr 233	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34		isufil 23291	. 2 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋) ↔ ({𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})))
35	6, 33, 34	sylanbrc 583	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4565  {csn 4591  ‘cfv 6501  (class class class)co 7362  fBascfbas 20821  filGencfg 20822  Filcfil 23233  UFilcufil 23287

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-fbas 20830  df-fg 20831  df-fil 23234  df-ufil 23289

This theorem is referenced by: (None)