Theorem fixufil 24048

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 24047	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
2	1	simprd 500	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))
3	1	simpld 499	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4		fgcl 24004	. . . 4 ⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
5	3, 4	syl 18	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
6	2, 5	eqeltrd 2869	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋))
7		undif2 4443	. . . . . . . . . 10 ⊢ (𝑦 ∪ (𝑋 ∖ 𝑦)) = (𝑦 ∪ 𝑋)
8		elpwi 4574	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋)
9		ssequn1 4147	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∪ 𝑋) = 𝑋)
10	8, 9	sylib 221	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝑦 ∪ 𝑋) = 𝑋)
11	7, 10	eqtr2id 2817	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑋 → 𝑋 = (𝑦 ∪ (𝑋 ∖ 𝑦)))
12	11	eleq2d 2855	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦))))
13	12	biimpac 483	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)))
14		elun 4115	. . . . . . 7 ⊢ (𝐴 ∈ (𝑦 ∪ (𝑋 ∖ 𝑦)) ↔ (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
15	13, 14	sylib 221	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
16	15	adantll 726	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ∨ 𝐴 ∈ (𝑋 ∖ 𝑦)))
17		ibar 537	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝑋 → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
18	17	adantl 486	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ 𝑦 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦)))
19		difss 4098	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝑦) ⊆ 𝑋
20		elpw2g 5304	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
21	19, 20	mpbiri 261	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
22	21	ad2antrr 738	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
23	22	biantrurd 541	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋 ∖ 𝑦) ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
24	18, 23	orbi12d 931	. . . . 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 235	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
26	25	ralrimiva 3163	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦) ∨ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦))))
27		eleq2 2858	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
28	27	elrab 3659	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑦))
29		eleq2 2858	. . . . . 6 ⊢ (𝑥 = (𝑋 ∖ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (𝑋 ∖ 𝑦)))
30	29	elrab 3659	. . . . 5 ⊢ ((𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ 𝐴 ∈ (𝑋 ∖ 𝑦)))
31	28, 30	orbi12i 927	. . . 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 3117	. . 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 237	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∨ (𝑋 ∖ 𝑦) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
34		isufil 24029	. 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 594	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4567  {csn 4594  ‘cfv 6537  (class class class)co 7411  fBascfbas 21479  filGencfg 21480  Filcfil 23971  UFilcufil 24025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21488  df-fg 21489  df-fil 23972  df-ufil 24027

This theorem is referenced by: (None)